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2002 AMC 10B Problem 10

Problem 10 of 25EasierAlgebra

Suppose that aa and bb are nonzero real numbers, and that the equation x2+ax+b=0x^2 + ax + b = 0 has solutions aa and b.b. What is the pair (a,b)?(a, b)?

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Solution

Since the roots are aa and b,b, Vieta’s formulas give a+b=−aa + b = -a and ab=b.ab = b. From ab=bab = b with b≠0,b \ne 0, we get a=1.a = 1. Then a+b=−aa + b = -a gives 1+b=−1,1 + b = -1, so b=−2.b = -2. Thus (a,b)=(1,−2),(a, b) = (1, -2), and the correct answer is C.
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Tagged: Vieta’s Formulas · system of equations

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