2000 AMC 10 Problems/Problem 15

Problem

Two non-zero real numbers, and , satisfy . Find a possible value of .

Solution

$\begin{align*} \frac{a}{b}+\frac{b}{a}-ab &= \frac{a^2+b^2}{ab}-ab \\ &= \frac{-a^2b^2+a^2+b^2}{ab}\\ \end(align*}$ (Error compiling LaTeX. Unknown error_msg)

Substituting , we get

$\begin{align*} \frac{-(a+b)^2+a^2+b^2}{ab} &= \frac{-a^2+2ab-b^2+a^2+b^2}{ab} \\ &= \frac{2ab}{ab} \\ &= 2 \\ \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Note: the answer is B. Check your calculations when you're substituting.