2008 Mock ARML 1 Problems/Problem 1

Problem

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

Solution

Let $f(x) = \sqrt{x+4}$; then $f(f(x)) = x$. Because $f(x)$ is increasing on $-4<x<\infty$, $f(f(x))=f(x)=x$. Using this we can show $x^2 - x - 4 = 0$. Using your favorite method, solve for $x = \frac{1 \pm \sqrt{17}}{2}$. However, since $f(x) =x$, and because the Square Root function's range does not include negative numbers, it follows that the negative root is extraneous, and thus we have $x = \boxed{\frac{1+\sqrt{17}}{2}}$.

See also

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

Square both sides twice leaving:

${x+4}=(x-4)^2$

Then, subtract $x-4$ to set to $0$ (from $x^2-8x^2+16$)

Using the rational roots theorem, we get the quadratics:

$(x^2-x-4)(x^2+x-3)$

Solve: $-1+-\sqrt{13}/2 1+-\sqrt{17}/2$

Seeing that negative roots are extraneous we have:

$1+\sqrt{17}/2$ and $-1+\sqrt{13}/2$ as the answers.

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