2018 AMC 12A Problem #5

Algebra Level 2

What is the sum of all possible values of k k for which the polynomials x 2 − 3 x + 2 x^2 - 3x + 2 and x 2 − 5 x + k x^2 - 5x + k have a root in common?

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4 6 5 3 10

Steven Yuan by Steven Yuan , 20, USA

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1 solution

Jerry McKenzie
Feb 9, 2018

Note x 2 − 3 x + 2 = ( x − 1 ) ( x − 2 ) x^2-3x+2=(x-1)(x-2) so if they have a root in common (if we call the missing root a), then by vieta's formula: ( x − 1 ) ( x − a ) = x 2 − 5 x + k ⇒ − 1 + − a = − 5 with a = k ⇒ − 1 + − k = − 5 ⇒ − k = − 4 ⇒ k = 4 (x-1)(x-a)=x^2-5x+k \Rightarrow -1+-a=-5 \enspace \textbf{with} \enspace a=k \Rightarrow -1+-k=-5 \Rightarrow -k=-4 \Rightarrow k=4

Likewise the other possibility: ( x − 2 ) ( x − a ) = x 2 − 5 x + k ⇒ k = 6 (x-2)(x-a)=x^2-5x+k \Rightarrow k=6

Thus the required answer is 4 + 6 = 10 4+6=10

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