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- Feb 14, 2012

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Given that $f(x)=x^2+12x+30$. Solve for the equation $f(f(f(f(f(x)))))=0$.

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- Thread starter
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- #1

- Feb 14, 2012

- 3,894

-----

Given that $f(x)=x^2+12x+30$. Solve for the equation $f(f(f(f(f(x)))))=0$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Feb 14, 2012

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1. castor28

2. MegaMoh

Partial credit goes to lfdahl for his partially correct solution!

Solution from castor28 :

The graph of $f$ has a minimum at $(-6,-6)$. This shows that $-6$ is a fixed point of $f$, and makes it worthwhile to try the substitution $x=z-6$. Under that substitution, we get:

$$

f(z-6) = z^2 - 6

$$

and, by induction,

$$

f^n(z-6) = z^{2^n}-6

$$

We must now solve:

$$

f^5(x) = f^5(z-6) = z^{32}-6 = 0

$$

giving the real roots $z=\pm\sqrt[32]{6}$ and $x=-6\pm\sqrt[32]{6}$.

The complex solutions are $x=-6 + \sqrt[32]{6}\,e^{\frac{2\pi ni}{32}}$, with $0\le n<32$.

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