There are exactly N distinct rational numbers k such that |k|<200 and 5x^2+kx+12=0 has at least one integer solution for x. What is N?

2014 AMC 12A Problem 19

Attempt Problem 19 of the 2014 AMC 12A 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 2014 AMC 12A solutions, or check the answer key.

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

19.

There are exactly

N

distinct rational numbers

k

such that

|k|\lt200

and

5x^2+kx+12=0

has at least one integer solution for

x.

What is

N?

6

12

24

48

78

x

is an integer root, solve for

k=-\dfrac{5x^2+12}{x}

=-\left(5x+\dfrac{12}{x}\right)

Require

|k|=5|x|+\dfrac{12}{|x|}\lt200

, and check the resulting values of

k

are all distinct

If an integer

x

is a root, then

k=-\left(5x+\dfrac{12}{x}\right),

x\ne0.

For

x\ge2,

|k|=5|x|+\dfrac{12}{|x|}

increases, and

|x|=39

gives

|k|\approx195.3\lt200,

while

|x|=40

gives

|k|\gt200.

Thus

x

ranges over

\pm1,\pm2,\dots,\pm39,

which is

78

values. If two integers

a\ne b

gave the same

k,

then

5a+\tfrac{12}{a}=5b+\tfrac{12}{b}

forces

5ab=12,

which has no integer solutions, so all

78

values of

k

are distinct.

Thus, the correct answer is E.