2015 AMC 12A Problem 18

Attempt Problem 18 of the 2015 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12A solutions, or check the answer key.

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18.

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

$7$

$8$

$16$

$17$

$18$

Answer: C

Concepts:Vieta’s Formulas Simon’s Favorite Factoring Trick

Difficulty rating: 1990

Solution:

Let the integer zeros be $p$ and $q.$ By Vieta's formulas $p + q = a$ and $pq = 2a,$ so $pq = 2(p+q).$ Rearranging gives $(p-2)(q-2) = 4.$

The integer factor pairs of $4$ are $(1,4),$ $(2,2),$ $(4,1),$ $(-1,-4),$ $(-2,-2),$ $(-4,-1),$ which yield $(p,q)$ pairs summing to $a = 9, 8, -1,$ and $0.$

The distinct possible values of $a$ are $9, 8, 0, -1,$ whose sum is $16.$

Thus, the correct answer is C.

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