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

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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1 Answer 1

up vote 8 down vote accepted

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?

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How do I prove that $z^2$ is real? –  Daniel Nov 3 '11 at 15:46
    
can you see any reason why the second and third term would be real? (taking the conjugate also helps, but this is simpler) –  N. S. Nov 3 '11 at 15:48
    
So finally it will be $4z^2+8z^2-3=0$? Thank you! –  Daniel Nov 3 '11 at 15:56
4  
Carefull, $z$ is complex, so $z^2$ can be also negative.... –  N. S. Nov 3 '11 at 16:00
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@nikita2 The last two are $\pm i \frac{\sqrt{3}}{2}$. –  N. S. Apr 5 '13 at 14:27

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