Difference between revisions of "1986 AIME Problems/Problem 1"
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== Solution == | == Solution == | ||
Let <math>y = \sqrt[4]{x}</math>. Then we have | Let <math>y = \sqrt[4]{x}</math>. Then we have | ||
| − | '''<math>y(7 - y) = 12</math>''', or, by simplifying, | + | '''<math>\displaystyle y(7 - y) = 12</math>''', or, by simplifying, |
| − | '''<math>y^2 - 7y + 12 = (y - 3)(y - 4) = 0</math>'''. | + | '''<math>\displaystyle y^2 - 7y + 12 = (y - 3)(y - 4) = 0</math>'''. |
| − | This means that <math>\sqrt[4]{x} = y = 3</math> or '''<math>4</math>'''. | + | This means that <math>\sqrt[4]{x} = y = 3</math> or '''<math>\displaystyle 4</math>'''. |
| − | Thus the sum of the possible solutions for '''<math>x</math>''' is '''<math>4^4 + 3^4 = 337</math>''', the answer. | + | Thus the sum of the possible solutions for '''<math>\displaystyle x</math>''' is '''<math>\displaystyle 4^4 + 3^4 = 337</math>''', the answer. |
== See also == | == See also == | ||
* [[1986 AIME Problems]] | * [[1986 AIME Problems]] | ||
Revision as of 19:51, 28 October 2006
Problem
What is the sum of the solutions to the equation ![$\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}$](http://latex.artofproblemsolving.com/f/4/1/f414e4ec0db945c286625cf56d2b6c7ede93f99d.png)
?
Solution
Let ![$y = \sqrt[4]{x}$](http://latex.artofproblemsolving.com/f/5/8/f5838ae6bb955f1f7afe1c28d870b6cd091b78ac.png)
. Then we have
, or, by simplifying,
.
This means that ![$\sqrt[4]{x} = y = 3$](http://latex.artofproblemsolving.com/7/2/3/723fc9924d37616a4675a26e831751451b6904d6.png)
or 
.
Thus the sum of the possible solutions for 
is 
, the answer.
