Difference between revisions of "2015 AMC 10A Problems/Problem 23"

Latest revision as of 10:55, 26 October 2025

Problem

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

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

Solution 1

By Vieta's Formula, $a$ is the sum of the integral zeros of the function, and so $a$ is integral.

Because the zeros are integral, the discriminant of the function, $a^2 - 8a$ , is a perfect square, say $k^2$ . Then adding 16 to both sides and completing the square yields $(a - 4)^2 = k^2 + 16.$ Therefore $(a-4)^2 - k^2 = 16$ and $((a-4) - k)((a-4) + k) = 16.$ Let $(a-4) - k = u$ and $(a-4) + k = v$ ; then, $a-4 = \dfrac{u+v}{2}$ and so $a = \dfrac{u+v}{2} + 4$ . Listing all possible $(u, v)$ pairs (not counting transpositions because this does not affect ( $u + v$ )), $(2, 8), (4, 4), (-2, -8), (-4, -4)$ , yields $a = 9, 8, -1, 0$ . These $a$ sum to $16$ , so our answer is $\boxed{\textbf{(C) }16}$ .

Solution 2

Let $r_1$ and $r_2$ be the integer zeroes of the quadratic. By Vieta's Formulas, $r_1 + r_2 = a\text{ and }r_1r_2 = 2a.$

Plugging the first equation in the second, $r_1r_2 = 2 (r_1 + r_2).$

Rearranging gives $r_1r_2 - 2r_1 - 2r_2 = 0 \implies (r_1 - 2)(r_2 - 2) = 4.$

These factors $(f_1,f_2)$ (ignoring order, because we want the sum of factors), can be $(1, 4), (-1, -4), (2, 2),$ or $(-2, -2)$ .

The sum of distinct $a = r_1 + r_2 = (f_1+2) + (f_2+2)$ , and these factors give $\sum_a a = (5+4) + (-5+4) + (4+4) + (-4+4) = \boxed{\textbf{(C) }16}$ .

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2015amc10a/397

Solution 3

Let the roots be $r$ and $s$ . By Vieta's we have that $r+s = a$ , and $rs = 2a$ . So by Simon's Favorite Factoring Trick: $2r+2s = rs \Longrightarrow rs-2r-2s = 0 \Longrightarrow (r-2)(s-2) = 4.$ Now, we test out values of $r$ and $s$ to see which work. Lets say that $r-2 = 1$ , and $s-2 = 4$ . This implies that $(r,s) = (3,6)$ , so $a = 9$ . Now, we let $r-2 = 2$ , and $s-2 = 2$ , this gives us $(r,s) = (4,4)$ , which gives us a value for $a$ of $8$ . Now, we circle an answer of $17$ and move onto the next problem. Yike! They didn't say that $a$ can't be negative! This is where many people would go wrong. Let $r-2 = -1$ , and $s-2 = -4$ . This gives us that $(r,s) = (1,-2)$ . This results in a value of $-1$ for $a$ . Now, we let $r-2 = -2$ , and $s-2 = -2$ . This gives $a = 0$ . Now, we are done. Our total sum is $9+8-1 = \boxed{\textbf{(C) }16}$ .

-jb2015007 - minor edits by mathlover1205 and PojoDotCom

Video Solution

https://youtu.be/RQ4ZCttwmA4

~savannahsolver

@@ Line 37: / Line 37: @@
 ==Solution 3==
-Let the roots be <math>r</math> and <math>s</math>. By Vieta's we have that <math>r+s = a</math>, and <math>rs = 2a</math>. So by Simon's Favorite Factoring Trick: <cmath>2r+2s = rs \Longrightarrow rs-2r-2s = 0 \Longrightarrow (r-2)(s-2) = 4</cmath>. Now, we test out values of <math>r</math> and <math>s</math> to see which work. Lets say that <math>r-2 = 1</math>, and <math>s-2 = 4</math>. This implies that <math>(r,s) = (3,6)</math>, so <math>a = 9</math>. Now, we let <math>r-2 = 2</math>, and <math>s-2 = 2</math>, this gives us <math>(r,s) = (4,4)</math>, which gives us a value for <math>a</math> of <math>8</math>. Now, we circle an answer of <math>17</math> and move onto the next problem. Yike! They didn't say that <math>a</math> can't be negative! This is where many people would go wrong. Let <math>r-2 = -1</math>, and <math>s-2 = -4</math>. This gives us that <math>(r,s) = (1,-2)</math>. This results in a value of <math>-1</math> for <math>a</math>. Now, we let <math>r-2 = -2</math>, and <math>s-2 = -2</math>. This gives <math>a = 0</math>. Now, we are done. Our total sum is <math>9+8-1 = \boxed{\textbf{(C) }16}</math>.
+Let the roots be <math>r</math> and <math>s</math>. By Vieta's we have that <math>r+s = a</math>, and <math>rs = 2a</math>. So by Simon's Favorite Factoring Trick: <cmath>2r+2s = rs \Longrightarrow rs-2r-2s = 0 \Longrightarrow (r-2)(s-2) = 4.</cmath> Now, we test out values of <math>r</math> and <math>s</math> to see which work. Lets say that <math>r-2 = 1</math>, and <math>s-2 = 4</math>. This implies that <math>(r,s) = (3,6)</math>, so <math>a = 9</math>. Now, we let <math>r-2 = 2</math>, and <math>s-2 = 2</math>, this gives us <math>(r,s) = (4,4)</math>, which gives us a value for <math>a</math> of <math>8</math>. Now, we circle an answer of <math>17</math> and move onto the next problem. Yike! They didn't say that <math>a</math> can't be negative! This is where many people would go wrong. Let <math>r-2 = -1</math>, and <math>s-2 = -4</math>. This gives us that <math>(r,s) = (1,-2)</math>. This results in a value of <math>-1</math> for <math>a</math>. Now, we let <math>r-2 = -2</math>, and <math>s-2 = -2</math>. This gives <math>a = 0</math>. Now, we are done. Our total sum is <math>9+8-1 = \boxed{\textbf{(C) }16}</math>.
 -jb2015007 - minor edits by mathlover1205 and PojoDotCom

2015 AMC 10A (Problems • Answer Key • Resources)
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