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2002 AMC 12A Problem 1

Problem 1 of 25EasierAlgebra

Compute the sum of all the roots of (2x+3)(x−4)+(2x+3)(x−6)=0. \begin{aligned} &(2x+3)(x-4) \\ &\quad {}+(2x+3)(x-6)=0. \end{aligned}

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Solution

Factoring out 2x+32x+3 gives (2x+3)[(x−4)+(x−6)](2x+3)\big[(x-4)+(x-6)\big] =(2x+3)(2x−10)= (2x+3)(2x-10) =0.= 0. The roots are −32-\dfrac{3}{2} and 5,5, whose sum is −32+5=72.-\dfrac{3}{2} + 5 = \dfrac{7}{2}. Thus, the correct answer is A.

More practice

Concepts: factoring · Vieta’s Formulas

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.