2006 AMC 10B Problem 14

Problem 14 of 25IntermediateAlgebra

Let

a

and

b

be the roots of the equation

x^2-mx+2=0.

Suppose that

a+\tfrac1b

and

b+\tfrac1a

are the roots of the equation

x^2-px+q=0.

What is

q?

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Solution

Since

a

and

b

are roots of

x^2-mx+2,

we have

ab=2.

The value

q

is the product of the new roots:

\begin{aligned} q&=\left(a+\tfrac1b\right)\left(b+\tfrac1a\right)\\ &=ab+1+1+\tfrac{1}{ab}\\ &=2+2+\tfrac12=\tfrac92. \end{aligned}

Thus, the correct answer is D.

More practice

Concepts: Vieta’s Formulas · symmetry (algebra)

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