Difference between revisions of "1986 AIME Problems/Problem 1"

Revision as of 18:51, 28 October 2006

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.

@@ Line 1: / Line 1: @@
 == Problem ==
-What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>?
+What is the sum of the solutions to the equation <math>\displaystyle \sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>?
 == Solution ==

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Difference between revisions of "1986 AIME Problems/Problem 1"

Revision as of 18:51, 28 October 2006

Problem

Solution

See also