2015 AMC 10A Problems/Problem 23

Problem

The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of the possible values of a?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}}\ 17\qquad\textbf{(E)}\ 18$ (Error compiling LaTeX. Unknown error_msg)

Solution

We use quadratic formula, yielding $x=\frac{a\pm \sqrt{a^2-8a}}{2}$ . We immediately see that $a^2-8a$ must be a perfect square in order for the solution to be rational. Thus, $a(a-8)$ is a perfect square. Note that if $a$ is more than $16$ or less than $-8$ , thus value cannot possibly be a perfect square. Trying all the values in between, $-1$ , $8$ , and $9$ work. Their sum yeilds $\boxed{\textbf{(C)}\ 16}$

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2015 AMC 10A Problems/Problem 23

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