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I have been trying to solve the following equation:

$z^2z^*−3z|z|+2z=0$

(z* is the conjugate of z)

I tried as following:

$z^2z^*−3z|z|+2z=0$

$z|z|^2−3z|z|+2z=0$

$z(|z|^2−3|z|+2)=0$

And from here I dont really know what I got to do (?). Also I tried to use Euler's formula and to substitute z with x+yi, but got stuck in the same way.

eojpyd
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    so either $z$ is identically zero or $|z|$ satisfies the above quadratic, constrain being $|z|$ now has to be positive – UnsinkableSam Jan 09 '22 at 15:25
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    "Also I tried to use Euler's formula and to substitute z with x+yi, but got stuck in the same way." Your analytical approach is very good, and your instincts about considering the substitutions of $z = (x + iy)$ or (perhaps) $z = re^{i\theta}$ is also good. As has been indicated by the other responses, your only stumble was not realizing that if $z$ does not happen to equal $[(0) + i(0)]$, then you can divide by $z$ to obtain the quadratic in $|z|$. – user2661923 Jan 09 '22 at 15:34
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    The last equation is $,z(|z|-1)(|z|-2)=0,$, and a product is $0$ iff one of the factors is $0$. – dxiv Jan 09 '22 at 21:05

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