# Solve for $z$ in $4z^2+8|z|^2-3=0$

$$4z^2+8|z|^2-3=0$$

I have to find $z$.

$|z|^2 = z\cdot \bar{z}$, but I don't know if this helps in this situation.

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You can prove first that $z^2$ is real, and then use the fact that $|z|^2=|z^2|$.
Can you find $z^2$ then?
How do I prove that $z^2$ is real? – Daniel Nov 3 '11 at 15:46
So finally it will be $4z^2+8z^2-3=0$? Thank you! – Daniel Nov 3 '11 at 15:56
Carefull, $z$ is complex, so $z^2$ can be also negative.... – N. S. Nov 3 '11 at 16:00
@nikita2 The last two are $\pm i \frac{\sqrt{3}}{2}$. – N. S. Apr 5 '13 at 14:27