1986 AIME Problems/Problem 1

Revision as of 14:43, 7 July 2008 by Joml88 (talk | contribs) (Solution)

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 $y(7 - y) = 12$, or, by simplifying, \[y^2 - 7y + 12 = (y - 3)(y - 4) = 0.\]

This means that $\sqrt[4]{x} = y = 3$ or $4$.

Thus the sum of the possible solutions for $x$ is $4^4 + 3^4 = 337$.

See also

1986 AIME (ProblemsAnswer KeyResources)
Preceded by
First Question
Followed by
Problem 2
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All AIME Problems and Solutions
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