# Number of solutions of a complex equation

Question: Find the number of solutions of the equation: $$z^3 + \frac{3(\bar z)^2}{|z|} = 0$$

I substituted $z = re^{i\theta}$ to convert the equation into: $$r^3e^{i3\theta} + 3re^{i2\theta}=0$$

This can be rearranged to be written as: $$re^{i2\theta}[r^2e^{i\theta} + 3] = 0$$

The only two solutions I see for this equation are $\theta = \pi$ and $r = \pm\sqrt3$. However, the answer given is 5. What have I dont wrong?

• $z^3 + \frac{3(\bar z)^2}{|z|}=r^3e^{i3\theta} + 3re^{-i2\theta}$ – πr8 May 14 '16 at 4:45
• @πr8 Aha... I'm an idiot.... Thanks. – Gummy bears May 14 '16 at 6:58