Equation with conjugate (complex numbers)

Question

I'm trying to solve this equation:

$(z-1)^3=9(\bar{z}-1)$

The problem is the conjugate.

$w=z-1$

Then:

$w^3=9(\bar{z}-1)$

Is there a relation between $\bar{z}$ and $w$?

Answer 1

Hint:

$$\overline z - 1 = \overline{z-1}$$

Answer 2

If $(z-1)^3=9(\overline z-1)$, then $\bigl|(z-1)^3\bigr|=9\bigl|\overline z-1\bigr|$, which means that $z=1$ or that $|z-1|^2=9$. So, either $z=1$ or $z=1+3(\cos\theta+i\sin\theta)$ for some $\theta$. Can you take it from here?