2015 AMC 10A Problems/Problem 23
Contents
Problem
The zeroes of the function 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. ! Extra }, or forgotten $.)
Solution 1
By Vieta's Formula, is the sum of the integral zeros of the function, and so is integral.
Because the zeros are integral, the discriminant of the function, , is a perfect square, say . Then adding 16 to both sides and completing the square yields Hence and Let and ; then, and so . Listing all possible pairs (not counting transpositions because this does not affect ), , yields . These sum to 16, so our answer is .
Solution 2
Let and be the integer zeroes of the quadratic.
Since the coefficent of the term is , the quadratic can be written as or .
By comparing this with , and .
Plugging the first equation in the second, . Rearranging gives .
This can be factored as .
These factors can be: .
We want the number of distinct , and these factors gives .
So the answer is .
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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