2022 AMC 12B Problem 4

Attempt Problem 4 of the 2022 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

If the roots are

p

and

q,

then

pq = 36

and

k = -(p+q)

Count the ordered-independent factor pairs of

36

with two distinct integers, including negatives

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.

Problem 4 in other years