Solve $(z-1)^3=9(\overline{z}-1)$ [duplicate]

Question

I would like to solve the complex equation $(z-1)^3=9(\overline{z}-1)$.

Here's what I did:

• Attempt 1: $w=z-1$, $w^3=9[cos(-\alpha)+isin(-\alpha)]-9$
• Attempt 2: $(x+iy-1)^3=9(x-iy-1)$
• Attempt 3: $z^3-3z^2+3z-1=9(\overline{z}-1)$

But it seems like neither of those attempts takes me to the solutions.

Answer 1

With $y=z-1$ you have $$y^3=9\bar y$$

Thus $$|y^3|=9|y|$$ We get $|y|=0$ or $|y|=3$

Thus $y=0$ or $y=3e^{i\theta }$

Plugging in $$y^3=9\bar y$$ gives us $e^{4i\theta}=1$

You can continue from there.