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<div class="section" id="fwk-redden-ch06_s01" condition="start-of-chunk" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">6.1</span> Extracting Square Roots and Completing the Square</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch06_s01_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch06_s01_o01" numeration="arabic">
<li>Solve certain quadratic equations by extracting square roots.</li>
<li>Solve any quadratic equation by completing the square.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch06_s01_s01" version="5.0" lang="en">
<h2 class="title editable block">Extracting Square Roots</h2>
<p class="para block" id="fwk-redden-ch06_s01_s01_p01">Recall that a quadratic equation is in <span class="margin_term"><a class="glossterm">standard form</a><span class="glossdef">Any quadratic equation in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0001" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span>, where <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0002" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span></span></span> if it is equal to 0:</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0003" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>
</span>
where <em class="emphasis">a</em>, <em class="emphasis">b</em>, and <em class="emphasis">c</em> are real numbers and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0004" display="inline"><mrow><mi>a</mi><mo>≠</mo><mn>0</mn></mrow><mo>.</mo></math></span> A solution to such an equation is a root of the quadratic function defined by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0005" display="inline"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>.</mo></math></span> Quadratic equations can have two real solutions, one real solution, or no real solution—in which case there will be two complex solutions. If the quadratic expression factors, then we can solve the equation by factoring. For example, we can solve <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0006" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span> by factoring as follows:</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0007" display="block"><mtable columnalign="left" columnspacing="0.1em"><mtr><mtd><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn><mo> </mo></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mrow><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p05">The two solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0008" display="inline"><mrow><mo>±</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>.</mo></math></span> Here we use ± to write the two solutions in a more compact form. The goal in this section is to develop an alternative method that can be used to easily solve equations where <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0009" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>0</mn></mrow></math></span>, giving the form</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0010" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p07">The equation <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0011" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span> is in this form and can be solved by first isolating <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0012" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0013" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p09">If we take the square root of both sides of this equation, we obtain the following:</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0014" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msqrt><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mo>|</mo><mi>x</mi><mo>|</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p11">Here we see that <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0015" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>±</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></math></span> are solutions to the resulting equation. In general, this describes the <span class="margin_term"><a class="glossterm">square root property</a><span class="glossdef">For any real number <em class="emphasis">k</em>, if <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0016" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>k</mi></mrow></math></span>, then <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0017" display="inline"><mrow><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mi>k</mi></msqrt></mrow><mo>.</mo></math></span></span></span>; for any real number <em class="emphasis">k</em>,</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0018" display="block"><mrow><mtext>if</mtext><mtext> </mtext><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>k</mi><mo>,</mo><mtext> </mtext><mtext>then</mtext><mtext> </mtext><mi>x</mi><mo>=</mo><mo>±</mo><msqrt><mi>k</mi></msqrt></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p13">Applying the square root property as a means of solving a quadratic equation is called <span class="margin_term"><a class="glossterm">extracting the root</a><span class="glossdef">Applying the square root property as a means of solving a quadratic equation.</span></span>. This method allows us to solve equations that do not factor.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p14">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0019" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p15">Notice that the quadratic expression on the left does not factor. However, it is in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0020" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span> and so we can solve it by extracting the roots. Begin by isolating <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0021" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0022" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>8</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>8</mn><mn>9</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p17">Next, apply the square root property. Remember to include the ± and simplify.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0023" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mfrac><mn>8</mn><mn>9</mn></mfrac></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p19">For completeness, check that these two real solutions solve the original quadratic equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p20">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2639" display="inline"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2640" display="inline"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2641" display="inline"><mtable><mtr><mtd columnalign="right"><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><msup><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><mo>⋅</mo><mn>2</mn></mrow><mn>9</mn></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2642" display="inline"><mtable><mtr><mtd columnalign="right"><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><msup><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>9</mn><mrow><mo>(</mo><mrow><mfrac><mrow><mn>4</mn><mo>⋅</mo><mn>2</mn></mrow><mn>9</mn></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>8</mn><mo>−</mo><mn>8</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch06_s01_s01_p21">Answer: Two real solutions, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0024" display="inline"><mrow><mo>±</mo><mfrac><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p22">Sometimes quadratic equations have no real solution. In this case, the solutions will be complex numbers.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p23">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0025" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p24">Begin by isolating <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0026" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and then apply the square root property.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0027" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>25</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>25</mn></mrow></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p26">After applying the square root property, we are left with the square root of a negative number. Therefore, there is no real solution to this equation; the solutions are complex. We can write these solutions in terms of the imaginary unit <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0028" display="inline"><mrow><mi>i</mi><mo>=</mo><msqrt><mrow><mo>−</mo><mn>1</mn></mrow></msqrt></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p27"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0029" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>25</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>1</mn><mo>⋅</mo><mn>25</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mi>i</mi><mo>⋅</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>5</mn><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p28">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2643" display="inline"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>5</mn><mi>i</mi></mrow></math></span></p></th>
<th align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2644" display="inline"><mrow><mi>C</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>5</mn><mi>i</mi></mrow></math></span></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2645" display="inline"><mtable><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mo>−</mo><mn>5</mn><mi>i</mi></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><msup><mi>i</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>25</mn><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch05_m2646" display="inline"><mtable><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mstyle color="#007fbf"><mn>5</mn><mi>i</mi></mstyle></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><msup><mi>i</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>25</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mo>−</mo><mn>25</mn><mo>+</mo><mn>25</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mn>0</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para" id="fwk-redden-ch06_s01_s01_p29">Answer: Two complex solutions, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0030" display="inline"><mrow><mo>±</mo><mn>5</mn><mi>i</mi></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p30"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0031" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p31">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0032" display="inline"><mrow><mo>±</mo><mfrac><mrow><msqrt><mn>6</mn></msqrt></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/9ff7QGhFytQ" condition="http://img.youtube.com/vi/9ff7QGhFytQ/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/9ff7QGhFytQ" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p33">Consider solving the following equation:</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p34"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0033" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p35">To solve this equation by factoring, first square <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0034" display="inline"><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span> and then put the equation in standard form, equal to zero, by subtracting 9 from both sides.</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p36"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0035" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p37">Factor and then apply the zero-product property.</p>
<p class="para block" id="fwk-redden-ch06_s01_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0036" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>8</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd><mtd><mrow></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd><mtd><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd><mtd><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p39">The two solutions are −8 and −2. When an equation is in this form, we can obtain the solutions in fewer steps by extracting the roots.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p40">Solve by extracting roots: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0037" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p41">The term with the square factor is isolated so we begin by applying the square root property.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0038" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>A</mi><mi>p</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>y</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mn>9</mn></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn><mo>±</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p43">At this point, separate the “plus or minus” into two equations and solve each individually.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p44"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0039" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn><mo>+</mo><mn>3</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>5</mn><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p45">Answer: The solutions are −2 and −8.</p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s01_p46">In addition to fewer steps, this method allows us to solve equations that do not factor.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p47">Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0040" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p48">Begin by isolating the term with the square factor.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p49"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0041" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>5</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p50">Next, extract the roots, solve for <em class="emphasis">x</em>, and then simplify.</p>
<p class="para" id="fwk-redden-ch06_s01_s01_p51"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0042" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>R</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>±</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt></mrow><mrow><mpadded height="1.2em"><msqrt><mn>2</mn></msqrt></mpadded></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mstyle color="#007fbf"><msqrt><mn>2</mn></msqrt></mstyle></mrow><mrow><mpadded height="1.2em"><mstyle color="#007fbf"><msqrt><mn>2</mn></msqrt></mstyle></mpadded></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>4</mn><mo>±</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p52">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0043" display="inline"><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0044" display="inline"><mrow><mfrac><mrow><mn>4</mn><mo>+</mo><msqrt><mrow><mn>10</mn></mrow></msqrt></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s01_s01_p53"><strong class="emphasis bold">Try this!</strong> Solve: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0045" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s01_p54">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0046" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mn>2</mn></mfrac><mi>i</mi></mrow><mo>.</mo></math></span></p>
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</div>
<div class="section" id="fwk-redden-ch06_s01_s02" version="5.0" lang="en">
<h2 class="title editable block">Completing the Square</h2>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p01">In this section, we will devise a method for rewriting any quadratic equation of the form</p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0047" display="block"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math>
</span>
as an equation of the form</p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0048" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mi>q</mi></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p05">This process is called <span class="margin_term"><a class="glossterm">completing the square</a><span class="glossdef">The process of rewriting a quadratic equation to be in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0049" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mi>q</mi></mrow><mo>.</mo></math></span></span></span>. As we have seen, quadratic equations in this form can be easily solved by extracting roots. We begin by examining perfect square trinomials:</p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0050" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>+</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mi> </mi><mn>6</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>+</mo></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi> </mi><mo>↓</mo></mstyle></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mo>↑</mo></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>6</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p07">The last term, 9, is the square of one-half of the coefficient of <em class="emphasis">x</em>. In general, this is true for any perfect square trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0051" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>.</mo></math></span></p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0052" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>⋅</mo><mfrac><mi>b</mi><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p09">In other words, any trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0053" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> will be a perfect square trinomial if</p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0054" display="block"><mrow><mi>c</mi><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p11"><strong class="emphasis bold">Note</strong>: It is important to point out that the leading coefficient must be equal to 1 for this to be true.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p12">Complete the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0055" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p13">In this example, the coefficient <em class="emphasis">b</em> of the middle term is −6. Find the value that completes the square as follows:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0056" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>6</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mn>9</mn></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p15">The value that completes the square is 9.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p16"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0057" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>9</mn></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p17">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0058" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>9</mn><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p18">Complete the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0059" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p19">Here <em class="emphasis">b</em> = 1. Find the value that will complete the square as follows:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p20"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0060" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p21">The value <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0061" display="inline"><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></math></span> completes the square:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p22"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0062" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p23">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0063" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p24">We can use this technique to solve quadratic equations. The idea is to take any quadratic equation in standard form and complete the square so that we can solve it by extracting roots. The following are general steps for solving a quadratic equation with leading coefficient 1 in standard form by completing the square.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n03">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p25">Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0064" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p26">It is important to notice that the leading coefficient is 1.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p27"><strong class="emphasis bold">Step 1</strong>: Add or subtract the constant term to obtain an equation of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0065" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>=</mo><mi>c</mi></mrow><mo>.</mo></math></span> Here we add 2 to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p28"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0066" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p29"><strong class="emphasis bold">Step 2</strong>: Use <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0067" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span> to determine the value that completes the square. In this case, <em class="emphasis">b</em> = −8:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p30"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0068" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>8</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mn>16</mn></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p31"><strong class="emphasis bold">Step 3</strong>: Add <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0069" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span> to both sides of the equation and complete the square.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p32"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0070" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>16</mn></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>16</mn></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>18</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>18</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p33"><strong class="emphasis bold">Step 4</strong>: Solve by extracting roots.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p34"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0071" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>18</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>18</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>±</mo><msqrt><mrow><mn>9</mn><mo>⋅</mo><mn>2</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>±</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p35">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0072" display="inline"><mrow><mn>4</mn><mo>−</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0073" display="inline"><mrow><mn>4</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mo>.</mo></math></span> The check is left to the reader.</p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n04">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p36">Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0074" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>48</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p37">Begin by adding 48 to both sides.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p38"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0075" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>48</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>48</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p39">Next, find the value that completes the square using <em class="emphasis">b</em> = 2.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p40"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0076" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mn>1</mn></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p41">To complete the square, add 1 to both sides, complete the square, and then solve by extracting the roots.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p42"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0077" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>48</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>C</mi><mi>o</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>s</mi><mi>q</mi><mi>u</mi><mi>a</mi><mi>r</mi><mi>e</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>1</mn></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>48</mn><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>1</mn></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>49</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>49</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>E</mi><mi>x</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mi>s</mi><mi>.</mi></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mn>49</mn></mrow></msqrt></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo>±</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p43">At this point, separate the “plus or minus” into two equations and solve each individually.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p44"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0078" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mn>7</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mtext>or</mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo>+</mo><mn>7</mn><mtext> </mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p45">Answer: The solutions are −8 and 6.</p>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p46"><strong class="emphasis bold">Note</strong>: In the previous example the solutions are integers. If this is the case, then the original equation will factor.</p>
<p class="para block" id="fwk-redden-ch06_s01_s02_p47"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0079" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>48</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p48">If an equation factors, we can solve it by factoring. However, not all quadratic equations will factor. Furthermore, equations often have complex solutions.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n05">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p49">Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0080" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>26</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p50">Begin by subtracting 26 from both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p51"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0081" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>26</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>26</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p52">Here <em class="emphasis">b</em> = −10, and we determine the value that completes the square as follows:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p53"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0082" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mo>−</mo><mn>10</mn></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mn>25</mn></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p54">To complete the square, add 25 to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p55"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0083" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>26</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>25</mn></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>26</mn><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>25</mn></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>25</mn></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p56">Factor and then solve by extracting roots.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p57"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0084" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mo>−</mo><mn>1</mn></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mi>i</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><mo>±</mo><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p58">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0085" display="inline"><mrow><mn>5</mn><mo>±</mo><mi>i</mi></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n05a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p59"><strong class="emphasis bold">Try this!</strong> Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0086" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>17</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p60">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0087" display="inline"><mrow><mi>x</mi><mo>=</mo><mn>1</mn><mo>±</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mo>.</mo></math></span></p>
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</div>
</div>
<p class="para editable block" id="fwk-redden-ch06_s01_s02_p62">The coefficient of <em class="emphasis">x</em> is not always divisible by 2.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n06">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p63">Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0088" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p64">Begin by subtracting 4 from both sides.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p65"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0089" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p66">Use <em class="emphasis">b</em> = 3 to find the value that completes the square:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p67"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0090" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mfrac><mn>9</mn><mn>4</mn></mfrac></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p68">To complete the square, add <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0091" display="inline"><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p69"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0092" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mfrac><mn>9</mn><mn>4</mn></mfrac></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>4</mn><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mfrac><mn>9</mn><mn>4</mn></mfrac></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>16</mn></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>9</mn><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>7</mn></mrow><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p70">Solve by extracting roots.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p71"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0093" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>7</mn><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mfrac><mrow><mo>−</mo><mn>1</mn><mo>⋅</mo><mn>7</mn></mrow><mn>4</mn></mfrac></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><mi>i</mi><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac><mi>i</mi></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p72">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0094" display="inline"><mrow><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mn>7</mn></msqrt></mrow><mn>2</mn></mfrac><mi>i</mi></mrow><mo>.</mo></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch06_s01_s02_p73">So far, all of the examples have had a leading coefficient of 1. The formula <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0095" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span> determines the value that completes the square only if the leading coefficient is 1. If this is not the case, then simply divide both sides by the leading coefficient before beginning the steps outlined for completing the square.</p>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n07">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p74">Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0096" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p75">Notice that the leading coefficient is 2. Therefore, divide both sides by 2 before beginning the steps required to solve by completing the square.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p76"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0097" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mstyle color="#007fbf"><mn>2</mn></mstyle></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>0</mn><mstyle color="#007fbf"><mn>2</mn></mstyle></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mn>2</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p77">Add <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0098" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p78"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0099" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>0</mn></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p79">Here <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0100" display="inline"><mrow><mi>b</mi><mo>=</mo><mfrac><mn>5</mn><mn>2</mn></mfrac></mrow></math></span>, and we can find the value that completes the square as follows:</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p80"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0101" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mstyle color="#007f3f"><mn>5</mn><mo>/</mo><mn>2</mn></mstyle></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>⋅</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p81">To complete the square, add <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0102" display="inline"><mrow><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></math></span> to both sides of the equation.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p82"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0103" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mstyle color="#007fbf"><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>8</mn><mrow><mn>16</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>25</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>33</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p83">Next, solve by extracting roots.</p>
<p class="para" id="fwk-redden-ch06_s01_s02_p84"><span class="informalequation"><math xml:id="fwk-redden-ch06_m0104" display="block"><mrow><mtable columnalign="left" columnspacing="0.1em"><mtr columnalign="left"><mtd columnalign="right"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>33</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><msqrt><mrow><mfrac><mrow><mn>33</mn></mrow><mrow><mn>16</mn></mrow></mfrac></mrow></msqrt></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>33</mn></mrow></msqrt></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mrow><mn>33</mn></mrow></msqrt></mrow><mn>4</mn></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mi>x</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><mo>±</mo><msqrt><mrow><mn>33</mn></mrow></msqrt></mrow><mn>4</mn></mfrac></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p85">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0105" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><mo>±</mo><msqrt><mrow><mn>33</mn></mrow></msqrt></mrow><mn>4</mn></mfrac></mrow><mo>.</mo></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch06_s01_s02_n07a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch06_s01_s02_p86"><strong class="emphasis bold">Try this!</strong> Solve by completing the square: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0106" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch06_s01_s02_p87">Answer: The solutions are <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0107" display="inline"><mrow><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>±</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow><mn>3</mn></mfrac><mi>i</mi></mrow><mo>.</mo></math></span></p>
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</div>
</div>
<div class="key_takeaways block" id="fwk-redden-ch06_s01_s02_n08">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch06_s01_s02_l01" mark="bullet">
<li>Solve equations of the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0108" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></mrow></math></span> by extracting the roots.</li>
<li>Extracting roots involves isolating the square and then applying the square root property. Remember to include “±” when taking the square root of both sides.</li>
<li>After applying the square root property, solve each of the resulting equations. Be sure to simplify all radical expressions and rationalize the denominator if necessary.</li>
<li>Solve any quadratic equation by completing the square.</li>
<li>You can apply the square root property to solve an equation if you can first convert the equation to the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0109" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mi>q</mi></mrow><mo>.</mo></math></span>
</li>
<li>To complete the square, first make sure the equation is in the form <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0110" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>=</mo><mi>c</mi></mrow><mo>.</mo></math></span> The leading coefficient must be 1. Then add the value <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0111" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>b</mi><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span> to both sides and factor.</li>
<li>The process for completing the square always works, but it may lead to some tedious calculations with fractions. This is the case when the middle term, <em class="emphasis">b</em>, is not divisible by 2.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch06_s01_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd01">
<h3 class="title">Part A: Extracting Square Roots</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch06_s01_qs01_p01"><strong class="emphasis bold">Solve by factoring and then solve by extracting roots. Check answers.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0112" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0113" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0114" display="inline"><mrow><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0116" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0118" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0119" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0120" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0122" display="inline"><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0124" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>25</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0125" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd01_qd02" start="11">
<p class="para" id="fwk-redden-ch06_s01_qs01_p22"><strong class="emphasis bold">Solve by extracting the roots.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0126" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>81</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p25"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0127" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0128" display="block"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>9</mn></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0130" display="block"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mn>16</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p31"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0132" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p33"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0134" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p35"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0136" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p37"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0138" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p39"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0140" display="inline"><mrow><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p41"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0142" display="inline"><mrow><mn>3</mn><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p43"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0144" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>40</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0146" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0148" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0150" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>100</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0152" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0154" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0156" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0158" display="inline"><mrow><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0160" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0162" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0164" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>4</mn><mn>9</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0166" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>9</mn><mrow><mn>25</mn></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0168" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0170" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0172" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0174" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>125</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0176" display="inline"><mrow><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0178" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0180" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0181" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>36</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0182" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>20</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p85"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0184" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>28</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0186" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>6</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0188" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0190" display="inline"><mrow><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>27</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0192" display="inline"><mrow><mn>9</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0194" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0196" display="inline"><mrow><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa49">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0198" display="block"><mrow><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa50">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0200" display="block"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0202" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>12</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0203" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p107">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0204" display="inline"><mrow><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>q</mi><mo>=</mo><mn>0</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0205" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p109">Solve for <em class="emphasis">x</em>: <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0207" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>p</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mi>q</mi><mo>=</mo><mn>0</mn></mrow></math></span>, <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0208" display="inline"><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>></mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p111">The diagonal of a square measures 3 centimeters. Find the length of each side.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p113">The length of a rectangle is twice its width. If the diagonal of the rectangle measures 10 meters, then find the dimensions of the rectangle.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p115">If a circle has an area of <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0213" display="inline"><mrow><mn>50</mn><mi>π</mi></mrow></math></span> square centimeters, then find its radius.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p117">If a square has an area of 27 square centimeters, then find the length of each side.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p119">The height in feet of an object dropped from an 18-foot stepladder is given by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0216" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>18</mn></mrow></math></span>, where <em class="emphasis">t</em> represents the time in seconds after the object is dropped. How long does it take the object to hit the ground? (Hint: The height is 0 when the object hits the ground. Round to the nearest hundredth of a second.)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p121">The height in feet of an object dropped from a 50-foot platform is given by <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0217" display="inline"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>16</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>50</mn></mrow></math></span>, where <em class="emphasis">t</em> represents the time in seconds after the object is dropped. How long does it take the object to hit the ground? (Round to the nearest hundredth of a second.)</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p123">How high does a 22-foot ladder reach if its base is 6 feet from the building on which it leans? Round to the nearest tenth of a foot.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p125">The height of a triangle is <span class="inlineequation"><math xml:id="fwk-redden-ch06_m0218" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></math></span> the length of its base. If the area of the triangle is 72 square meters, find the exact length of the triangle’s base.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd02">
<h3 class="title">Part B: Completing the Square</h3>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd02_qd01" start="63">
<p class="para" id="fwk-redden-ch06_s01_qs01_p127"><strong class="emphasis bold">Complete the square.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0220" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0222" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0224" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0226" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0228" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0230" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0232" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0234" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p144"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0236" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p146"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0238" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mtext> </mtext><mtext> </mtext><mo>?</mo><mtext> </mtext></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd02_qd02" start="73">
<p class="para" id="fwk-redden-ch06_s01_qs01_p148"><strong class="emphasis bold">Solve by factoring and then solve by completing the square. Check answers.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0240" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0241" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>15</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p153"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0242" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>24</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0243" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>y</mi><mo>+</mo><mn>11</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0244" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>t</mi><mo>−</mo><mn>28</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p159"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0245" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>t</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p161"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0246" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p163"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0248" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p165"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0250" display="inline"><mrow><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mi>y</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p167"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0252" display="inline"><mrow><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>y</mi><mo>−</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd02_qd03" start="83">
<p class="para" id="fwk-redden-ch06_s01_qs01_p169"><strong class="emphasis bold">Solve by completing the square.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0254" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0256" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0258" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>7</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0260" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0262" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0264" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>9</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0266" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>t</mi><mo>−</mo><mn>75</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0267" display="inline"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>t</mi><mo>−</mo><mn>108</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0268" display="block"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>u</mi><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0270" display="block"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mi>u</mi><mo>−</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0272" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0274" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0276" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>y</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0278" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>y</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0280" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0282" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa99">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0284" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mfrac><mrow><mn>11</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa100">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0286" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0288" display="block"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>t</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch06_m0290" display="block"><mrow><msup><mi>t</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>t</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa103">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p210"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0292" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa104">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p212"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0294" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p214"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0296" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p216"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0298" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p218"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0300" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p220"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0302" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p222"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0304" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>15</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p224"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0306" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>43</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa111">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p226"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0308" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>10</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa112">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p228"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0310" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>x</mi><mo>+</mo><mn>42</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa113">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p230"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0312" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa114">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p232"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0314" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa115">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p234"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0316" display="inline"><mrow><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>u</mi><mo>−</mo><mn>2</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa116">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p236"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0318" display="inline"><mrow><mn>3</mn><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mi>u</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa117">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p238"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0320" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>=</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa118">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p240"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0322" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>8</mn><mo>=</mo><mo>−</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa119">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p242"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0324" display="inline"><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>11</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa120">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p244"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0326" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>4</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa121">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p246"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0328" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa122">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p248"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0330" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mi>y</mi><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa123">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p250"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0332" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa124">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p252"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0334" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>t</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>t</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mn>10</mn><mi>t</mi></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch06_s01_qs01_qd02_qd04" start="125">
<p class="para" id="fwk-redden-ch06_s01_qs01_p254"><strong class="emphasis bold">Solve by completing the square and round the solutions to the nearest hundredth.</strong></p>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa125">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p255"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0336" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa126">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p257"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0337" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><mn>5</mn><mo>−</mo><mn>15</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch06_s01_qs01_qa127">
<div class="question">
<p class="para" id="fwk-redden-ch06_s01_qs01_p259"><span class="inlineequation"><math xml:id="fwk-redden-ch06_m0338" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>