# Difference between revisions of "2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2"

## Problem

If $f(x) = x^3 + 6x^2 + 12x + 6$, solve the equation $f(f(f(x))) = 0.$

## Solution

We can write $f(x) = x^3 + 6x^2 + 12x + 6 = (x+2)^3 -2$. This means that $f(x) = 0 \Rightarrow x = \sqrt[3]{2} - 2$.

Working backwards, $f(f(f(x))) = 0 \Rightarrow f(f(x)) = \sqrt[3]{2} - 2$.

Then, $f(f(x)) = (f(x)-2)^3 - 2 = \sqrt[3]{2} - 2 \Rightarrow f(x) = \sqrt[9]{2}$.

Finally, $f(x) = (x-2)^3 - 2 = \sqrt[9]{2} \Rightarrow \sqrt[3]{2+\sqrt[9]{2}}$.