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2008 Mock ARML 1 Problems/Problem 1

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Problem

Compute all real values of x such that \sqrt {\sqrt {x + 4} + 4} = x.

Solution

Let f(x) = \sqrt{x+4}; then f(f(x)) = x. Suppose that f(x) = x \Longleftrightarrow x^2 - x - 4 = 0 \Longrightarrow x = \frac{1 \pm \sqrt{17}}{2}. However, since f(x) > 0, it follows that the negative root is extraneous, and thus we have x = \boxed{\frac{1+\sqrt{17}}{2}}. The other roots we can verify are not real. This solution is incomplete. You can help us out by completing it.

See also

2008 Mock ARML 1 (Problems, Source)
Preceded by
First question
Followed by
Problem 2
1 2 3 4 5 6 7 8
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