2014 AMC 12A 第 19 题

If $x$ is an integer root, solve for $k=-\dfrac{5x^2+12}{x}$ $=-\left(5x+\dfrac{12}{x}\right)$

Require $|k|=5|x|+\dfrac{12}{|x|}\lt200$, and check the resulting values of $k$ are all distinct

If an integer $x$ is a root, then $k=-\left(5x+\dfrac{12}{x}\right),$ so $x\ne0.$ For $x\ge2,$ $|k|=5|x|+\dfrac{12}{|x|}$ increases, and $|x|=39$ gives $|k|\approx195.3\lt200,$ while $|x|=40$ gives $|k|\gt200.$

Thus $x$ ranges over $\pm1,\pm2,\dots,\pm39,$ which is $78$ values. If two integers $a\ne b$ gave the same $k,$ then $5a+\tfrac{12}{a}=5b+\tfrac{12}{b}$ forces $5ab=12,$ which has no integer solutions, so all $78$ values of $k$ are distinct.

Thus, the correct answer is E.