how to simplify and solve this equation :

Question

Solve this equation for $x$:

$$\left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}$$

I'm having a problem getting rid of the square root!

Answer 1

Provided $-1\le x\le 1$, you get:

\begin{align} \left|x \sqrt{1-x^2} +x \right| = \sqrt{1+x^2}\\ \left|x \left(\sqrt{1-x^2} +1\right) \right| = \sqrt{1+x^2}\\ |x|\left(\sqrt{1-x^2} +1\right) = \sqrt{1+x^2}\\ x^2\left(\sqrt{1-x^2} +1\right)^2 = 1+x^2\\ x^2\left(1-x^2+2\sqrt{1-x^2}+1\right) = 1+x^2\\ 2x^2\sqrt{1-x^2}=x^4-x^2+1\\ 4x^4\left(1-x^2\right)=\left(x^4-x^2+1\right)^2 \end{align}