1991 AIME Problems/Problem 8
Problem
For how many real numbers does the quadratic equation
have only integer roots for
?
Contents
Solution
By Vieta's formulas, where
are the roots of the quadratic, and since
are integers,
must be an integer. Applying the quadratic formula,
Since is an integer, we need
to be an integer (let this be
):
. Completing the square, we get
Which implies that is a perfect square also (let this be
). Then
The pairs of factors of are
; since
is the average of each respective pair and is also an integer, the pairs that work must have the same parity. Thus we get
pairs (counting positive and negative) of factors that work, and substituting them backwards show that they all work.
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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