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Sum of possible Ks

Source: 2018 AMC 12A #5

February 8, 2018
2018 AMC 12AAMCAMC 12AMC 12 Aalgebrapolynomial

Problem Statement

What is the sum of all possible values of kk for which the polynomials x23x+2x^2 - 3x + 2 and x25x+kx^2 - 5x + k have a root in common?
<spanclass=latexbold>(A)</span>3<spanclass=latexbold>(B)</span>4<spanclass=latexbold>(C)</span>5<spanclass=latexbold>(D)</span>6<spanclass=latexbold>(E)</span>10 <span class='latex-bold'>(A) </span>3 \qquad <span class='latex-bold'>(B) </span>4 \qquad <span class='latex-bold'>(C) </span>5 \qquad <span class='latex-bold'>(D) </span>6 \qquad <span class='latex-bold'>(E) </span>10 \qquad