Solve equation with a conjugate How to solve this equation:
$$
|z|^2 - 9 - \bar z + 3i = 0
$$
I know that:
$$
|z|^2=z \bar z
$$
$$
|z|=\sqrt{a^2+b^2}
$$
$$
z = a + bi \Rightarrow \bar z = a - bi
$$
But I still don't know to calculate z from that.

Be $z=x+iy$ so $|z|^2=x^2+y^2$, $\bar z=x-iy$. Replacing in the equation:$$|z|^2-9-\bar z+3i=0$$ $$x^2+y^2-9-x+iy+3i=0$$
Reagrouping,$$(x^2+y^2-9-x)+i(y+3)=0$$
So $Re(z)=0$ and $Im(z)=0$,$$y+3=0$$$$x^2+y^2-9-x=0$$
Solving for $y$ gives $y=-3$. Replacing in the other equation:
$$x^2+(-3)^2-9-x=0$$
$$x^2-x=0$$
Gives solutions to $x$, $x=0$ $x=1$.
So you have to solutions for z.$z=-3i$ and $z=1-3i$
