Solve the equation
$$\left( x^2+100 \right)^2 = \left( x^3 -100 \right)^3$$
I have no idea what to do. The equation is part of an exercise that goes as follows:
Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that
$$f(f(x)) = x \quad \forall \; x \in \mathbb{R}$$
- Prove that $f$ is $1-1$ or a bijection. ( done )
- If $f$ is strictly increasing , then prove that $f(x)=x$. ( done )
- Solve the equation $\left( x^2+100 \right)^2 = \left( x^3 -100 \right)^3$ using the above.
No idea how to combine the above to solve the equation. I'm interested in solutions that use high school methods preferably no derivatives; just pure functions (e.g composition or simple algebraic manipulations )
Wolfram Alpha gives $x=5$ as a solution.
