Can't solve complex equation

Question

Find all $z$ satisfying:

$$e^z-2ie^{-z}=i-2$$

I jsut don't have any idea how can one solve it in a simple way. Please help.

Answer 1

$\textbf{Hint:}$ Multiply both sides by $e^{z}$:

$$e^{2z}-2i=(i-2)e^{z}$$

Let $x=e^{z}$

$$x^2-2i=(i-2)x$$

It's quadratic equation.