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

Revision as of 13:43, 7 July 2008

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

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 $4^4 + 3^4 = 337$ .

@@ Line 4: / Line 4: @@
 == Solution ==
 Let <math>y = \sqrt[4]{x}</math>. Then we have
-'''<math>\displaystyle y(7 - y) = 12</math>''', or, by simplifying,
+'''<math>y(7 - y) = 12</math>''', or, by simplifying,
-'''<math>\displaystyle y^2 - 7y + 12 = (y - 3)(y - 4) = 0</math>'''.
+'''<cmath>y^2 - 7y + 12 = (y - 3)(y - 4) = 0.</cmath>'''
-This means that <math>\sqrt[4]{x} = y = 3</math> or '''<math>\displaystyle 4</math>'''.
-Thus the sum of the possible solutions for '''<math>\displaystyle x</math>''' is '''<math>\displaystyle 4^4 + 3^4 = 337</math>''', the answer.
+This means that <math>\sqrt[4]{x} = y = 3</math> or '''<math>4</math>'''.
+Thus the sum of the possible solutions for '''<math>x</math>''' is '''<math>4^4 + 3^4 = 337</math>'''.
 == See also ==

1986 AIME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions