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2015 AMC 12B Problem 12

Problem 12 of 25IntermediateAlgebra

Let a,a, b,b, and cc be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation (x−a)(x−b)(x - a)(x - b) +(x−b)(x−c)=0?+ (x - b)(x - c) = 0?

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Solution

Factoring gives (x−b)(2x−(a+c))=0,(x - b)\bigl(2x - (a + c)\bigr) = 0, so the roots are bb and a+c2.\dfrac{a + c}{2}. Their sum is b+a+c2.b + \dfrac{a + c}{2}. Using distinct digits, take b=9b = 9 and a+c=8+7=15,a + c = 8 + 7 = 15, giving 9+7.5=16.5.9 + 7.5 = 16.5. Thus, the correct answer is D.

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Concepts: factoring · optimization

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