2002 AMC 12A Problem 13

Problem 13 of 25IntermediateAlgebra

Two different positive numbers

a

and

b

each differ from their reciprocals by

1.

What is

a + b?

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Solution

A positive number

x

differs from its reciprocal by

1

when

x - \dfrac1x = 1

or

x - \dfrac1x = -1,

i.e.

x^2 - x - 1 = 0

or

x^2 + x - 1 = 0.

The positive roots are

\dfrac{1+\sqrt5}{2}

and

\dfrac{-1+\sqrt5}{2},

which are reciprocals of each other. Their sum is

a + b = \sqrt5.

Thus, the correct answer is C.

More practice

Concepts: rational equation · quadratic

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