2015 AMC 12A Problems/Problem 18

Problem

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

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 17 \qquad\textbf{(E)}\ 18$

Solution 1

The problem asks us to find the sum of every integer value of $a$ such that the roots of $x^2 - ax + 2a = 0$ are both integers.

The quadratic formula gives the roots of the quadratic equation: $x=\frac{a\pm\sqrt{a^2-8a}}{2}$

As long as the numerator is an even integer, the roots are both integers. But first of all, the radical term in the numerator needs to be an integer; that is, the discriminant $a^2 - 8a$ equals $k^2$ , for some nonnegative integer $k$ .

$a^2-8a=k^2$

$a(a-8)=k^2$

$((a-4)+4)((a-4)-4)=k^2$

$(a-4)^2-4^2=k^2$

$(a-4)^2=k^2+4^2$

From this last equation, we are given a hint of the Pythagorean theorem. Thus, $(k,4,|a-4|)$ must be a Pythagorean triple unless $k = 0$ .

In the case $k=0$ , the equation simplifies to $|a-4|=4$ . From this equation, we have $a=0,8$ . For both $a=0$ and $a=8$ , $\frac{a\pm\sqrt{a^2-8a}}{2}$ yields two integers, so these values satisfy the constraints from the original problem statement. (Note: the two zero roots count as "two integers.")

If $k$ is a positive integer, then only one Pythagorean triple could match the triple $(k,4,|a - 4|)$ because the only Pythagorean triple with a $4$ as one of the values is the classic $(3,4,5)$ triple. Here, $k=3$ and $|a-4|=5$ . Hence, $a=-1,9$ . Again, $\frac{a\pm\sqrt{a^2-8a}}{2}$ yields two integers for both $a=-1$ and $a=9$ , so these two values also satisfy the original constraints.

There are a total of four possible values for $a$ : $-1,0,8,$ and $9$ . Hence, the sum of all of the possible values of $a$ is $\boxed{\textbf{(C) }16}$ .

Solution 2

By the quadratic formula, the roots $r$ can be represented by $r=\frac{a\pm\sqrt{a^2-8a}}{2}$ For $r\in\mathbb{Z}$ , $a\in\mathbb{Z}$ , since $\frac{\sqrt{a^2-8a}}{2}$ and $\frac{a}{2}$ will have different mantissas (mantissae?).

Now observe the discriminant $\sqrt{a^2-8a}=\sqrt{a(a-8)}$ and have two cases.

Positive $a$

$a\geq8$ and $a\leq0$ , since $1\geq a \geq7$ gives imaginary roots. Testing positive $a$ values, quickly see that $a\leq9$ . After $16$ and $36$ , the difference between the closest nonzero factor pairs of perfect squares exceeds $8$ . For $8\geq a \geq9$ , $a=8,9$ . Checking both yields an integer.

Negative $a$

We can instead test with $\sqrt{-a(8-a)}$ . If $b=8-a$ , we have our original expression. Thus, for the same reasons, $b=8,9\implies 8,9=8-a$ . $a=-1$ (0 does not affect the answer).

$-1+8+9=16\implies\boxed{\textbf{(C) }16}$

(Solution by BJHHar)

Solution 3

Let $m$ and $n$ be the roots of $x^2-ax+2a$

By Vieta's Formulas, $n+m=a$ and $mn=2a$

Substituting gets us $n+m=\frac{mn}{2}$

$2n-mn+2m=0$

Using Simon's Favorite Factoring Trick:

$n(2-m)+2m=0$

$-n(2-m)-2m=0$

$-n(2-m)-2m+4=4$

$(2-n)(2-m)=4$

This means that the values for $(m,n)$ are $(0,0),(4,4),(3,6),(1,-2)$ giving us $a$ values of $-1,0,8,$ and $9$ . Adding these up gets $\boxed{\textbf{(C) }16}$ .

Solution 4

The quadratic formula gives $x = \frac{a \pm \sqrt{a(a-8)}}{2}$ . For $x$ to be an integer, it is necessary (and sufficient!) that $a(a-8)$ to be a perfect square. So we have $a(a-8) = b^2$ ; this is a quadratic in itself and the quadratic formula gives $a = 4 \pm \sqrt{16 + b^2}$

We want $16 + b^2$ to be a perfect square. From smartly trying small values of $b$ , we find $b = 0, b = 3$ as solutions, which correspond to $a = -1, 0, 8, 9$ . These are the only ones; if we want to make sure then we must hand check up to $b=8$ . Indeed, for $b \geq 9$ we have that the differences between consecutive squares are greater than $16$ so we can't have $b^2 + 16$ be a perfect square. So summing our values for $a$ we find $\boxed{16 \textbf{ (C)}}$ . as the answer.

Additional note: You can use the quadratic and plug in squares for a (since for $b^2$ to be an integer $a$ would have to be some square), and eventually you can notice a limit to get the answer~

Solution 5

First of all, we know that $a$ is the sum of the quadratic's two roots, by Vieta's formulas. Thus, $a$ must be an integer. Then, we notice that the discriminant $a^2-8a$ must be equal to a perfect square so that the roots are integers. Thus, $a(a-8)=b^2$ where $b$ is an integer.

We can complete the square and rearrange to get $(a-4)^2-b^2=16$ . Let's define $m=a-4$ , just to make things a little easier to write, so now we have $(m+b)(m-b)=16$ . We can now list out the integer factor pairs of 16 and the resulting values of $m$ and $b$ . (Note that $m$ and $b$ must both be integers)

$(m+b)(m-b)=16$

$16*1=16$ $\Rightarrow$ Doesn't work

$8*2=16$ $\Rightarrow$ $m=5, b=3$

$4*4=16$ $\Rightarrow$ $m=4, b=0$

$2*8=16$ $\Rightarrow$ $m=5, b=-3$

$1*16=16$ $\Rightarrow$ Doesn't work

$-16*-1=16$ $\Rightarrow$ Doesn't work

$-8*-2=16$ $\Rightarrow$ $m=-5, b=-3$

$-4*-4=16$ $\Rightarrow$ $m=-4, b=0$

$-2*-8=16$ $\Rightarrow$ $m=-5, b=-3$

$-1*-16=16$ $\Rightarrow$ Doesn't work

We want the possible values of $m$ , which are $-5,-4,4,$ and $5$ . As $m+4=a$ , $a$ can equal $-1,0,8,$ or $9.$ Adding all of that up gets us our answer, $\boxed{\textbf{(C) }16}$ .

(Solution by Curious_crow)

Solution 6 (big fast)

Let our roots be $r$ and $s$ . By vieta's formulas, we have that $r+s=a$ and $rs = 2a$ . Thus, $rs = 2r+2s$ , or $(r-2)(s-2) = 4$ .

4 can only be obtained from (2,2), (1,4), (-2,-2), or (-1,-4), giving us $-1,0,8$ and $9$ , which add up to $\boxed{\textbf{(C)}16}$

-skibbysiggy

2015 AMC 12A (Problems • Answer Key • Resources)
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