1986 AIME Problems/Problem 1

Problem

What is the sum of the solutions to the equation $\displaystyle \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$ ?

Solution

Let $y = \sqrt[4]{x}$ . Then we have $\displaystyle y(7 - y) = 12$ , or, by simplifying, $\displaystyle y^2 - 7y + 12 = (y - 3)(y - 4) = 0$ . This means that $\sqrt[4]{x} = y = 3$ or $\displaystyle 4$ . Thus the sum of the possible solutions for $\displaystyle x$ is $\displaystyle 4^4 + 3^4 = 337$ , the answer.

Page

Toolbox

Search

1986 AIME Problems/Problem 1

Problem

Solution

See also