What is the sum of the solutions to the equation √[4]x=127-√[4]x?

1986 AIME Problem 1

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What is the sum of the solutions to the equation

\sqrt[4]{x}=\frac{12}{7-\sqrt[4]{x}}?

Answer

Substitute

t=\sqrt[4]{x}

After clearing the denominator, factor the resulting quadratic in

t

Let

t=\sqrt[4]{x},

t\geq0.

The equation becomes

t(7-t)=12,

t^2-7t+12=0.

Thus

t=3

t=4,

and both values are valid in the original equation. Hence

x=3^4=81

x=4^4=256,

and their sum is

81+256=337.