2018 AMC 12A Problem 5

Below is the professionally curated solution for Problem 5 of the 2018 AMC 12A, 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.

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Concepts:quadratic factoring

Difficulty rating: 1270

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?

$3$

$4$

$5$

$6$

$10$

Solution:

Since $x^2 - 3x + 2 = (x-1)(x-2),$ its roots are $1$ and $2.$ If $1$ is a shared root then $1 - 5 + k = 0,$ so $k = 4.$ If $2$ is a shared root then $4 - 10 + k = 0,$ so $k = 6.$ The sum of possible values is $4 + 6 = 10.$

Thus, the correct answer is E.

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