2000 AMC 10 Problem 15

Problem 15 of 25IntermediateAlgebra

Two non-zero real numbers,

a

and

b,

satisfy

ab = a - b.

Find a possible value of

\dfrac{a}{b} + \dfrac{b}{a} - ab.

Show solution

Solution

Over the common denominator

ab,

\dfrac{a}{b} + \dfrac{b}{a} - ab = \dfrac{a^2 + b^2 - (ab)^2}{ab}.

Substituting

ab = a - b

gives

\dfrac{a^2 + b^2 - (a - b)^2}{ab} = \dfrac{2ab}{ab} = 2.

Thus, the correct answer is E.

Concepts: algebraic manipulation · substitution

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