idk

academic/misc/2024-math-calendar/12-dec.html

@@ -970,8 +970,33 @@ <h3 id="dec18">Dec 18</h3>
 
 <h3 id="dec19">Dec 19</h3>
 
+<p>
+Find the larger solution to \(4x^2-36x=x^2+20x+19\)
+</p>
+
+<p>
+\[3x^2-56x-19=0\]
+By factoring
+\[(3x+1)(x-19)=0\Rightarrow x=-1/3,19\]
+By quadratic formula
+\[\begin{align}&x={56\pm\sqrt{56^2+4\cdot3\cdot19}\over6}
+={56\pm\sqrt{56^2+2\cdot2\cdot56+2^2}\over6}\\
+&={56\pm\sqrt{(56+2)^2}\over6}={56\pm58\over6}=-1/3,19\end{align}\]
+</p>
+
 <h3 id="dec20">Dec 20</h3>
 
+<p>
+How many times does 2 appear in the natural numbers under 100?
+</p>
+
+<p>
+In single digits, it is just the number 2. In double digits, there are 8 numbers
+with a tens digit other than 2 and a ones digit of 2. If the tens digit is 2,
+then 2 appears 10 times as the tens digit and once as the ones digit. In total,
+there are 20 appearances.
+</p>
+
 <h3 id="dec21">Dec 21</h3>
 
 <h3 id="dec22">Dec 22</h3>