Complex Numbers: How to find the number of solutions of $z^3 + \overline{z} = 0$

Question

Find the number of solutions of $z^3+ \overline{z}=0$.

I tried to write $z=x+iy$ and then expand $z^3$, but I am not getting anything from it.

Please help me out.

Answer 1

There is a better way than writing $z = x + iy$ and expanding.

Follow these steps/hints:

1. Rewrite the equation as $z^3 = -\overline{z}$. What can you say about $|z|$?

2. In view of 1., how can you express $\overline{z}$ in terms of $z$?

3. Now how does the equation look like?

Answer 2

$z^3+ \overline{z}=0$ implies $|z|^3=|z|$ and so $z=0$ or $|z|=1$.

If $|z|=1$, then $\overline{z}=z^{-1}$ and so $z^3=-z^{-1}$. Thus $z^4=-1$.

Can you take it from here?