Well, to solve it I firstly tried to use exponential form. However, it didn't give me expected result. Also, I tried using trigonometric form but I am a bit inexperienced in it, so I couldn't observe anything interesting with it either.
I am new to complex numbers, so I would love to hear some advices on how to think of solution in such equations and maybe some insights on how to solve this one.
Best Answer
Note that\begin{align}\require{cancel}x^2|x|=2\overline x&\implies\bigl|x^2|x|\bigr|=\left|2\overline x\right|\\&\iff|x|^3=2|x|.\end{align}Therefore, $|x|=0$ or $|x|=\sqrt2$. In the first case, we have $x=0$, which is indeed a solution of the equation. Otherwise, $x=\sqrt2e^{i\theta}$, for some $\theta\in\Bbb R$, and your equation becomes$$\cancel{2\sqrt2}e^{2i\theta}=\cancel{2\sqrt2}e^{-i\theta}.$$Can you take it from here?