How to solve $z^2\overline z-z\overline z=-\overline z$

Question

I was trying to solve the equation using the identities $z=x+iy$; $\overline z=x-iy$ and $z\overline z=x^2+y^2$, but I came up with a tougher one. Looking at the solution it says that if $z=0$, obviuosly the equation is satisfied, but if $z\not=0$ then also $\overline z \not=0$, and the equation is equivalent to a quadratic one $z^2-z+1=0$. I can't understand how to reach that result.

Answer 1

We have

\begin{align*} z^2 \bar{z} - z \bar{z} & = - \bar{z} \quad \text{divide by } \bar{z} \neq0\\ z^2 - z & = -1 \\ z^2 - z +1 & = 0 \end{align*}