2018 AMC 12A Problem 5

Attempt Problem 5 of the 2018 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 2018 AMC 12A solutions, or check the answer key.

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

What is the sum of all possible values of

k

for which the polynomials

x^2 - 3x + 2

and

x^2 - 5x + k

have a root in common?

3

4

5

6

10

Factor

x^2 - 3x + 2

to find its two roots

A common root must be

1

2;

substitute each into

x^2 - 5x + k

and solve for

k

Since

x^2 - 3x + 2 = (x-1)(x-2),

its roots are

1

and

2.

1

is a shared root then

1 - 5 + k = 0,

k = 4.

2

is a shared root then

4 - 10 + k = 0,

k = 6.

The sum of possible values is

4 + 6 = 10.

Thus, the correct answer is E.

Problem 5 in other years