2022 AMC 12B Problem 4

Below is the professionally curated solution for Problem 4 of the 2022 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 12B solutions, or check the answer key.

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Concepts:Vieta’s Formulas factor systematic listing

Difficulty rating: 1200

For how many values of the constant $k$ will the polynomial $x^2 + kx + 36$ have two distinct integer roots?

$6$

$8$

$9$

$14$

$16$

Solution:

If the roots are integers $p$ and $q,$ then $pq = 36$ and $k = -(p+q).$ Distinct roots must have the same sign, so we list factor pairs of $36$ with $p \ne q.$

The positive pairs are $(1,36), (2,18), (3,12), (4,9),$ and the negative pairs are $(-1,-36), (-2,-18), (-3,-12), (-4,-9).$ The pair $(6,6)$ is excluded since the roots must be distinct.

Each of these $8$ pairs gives a different value of $k.$

Thus, the correct answer is B.

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