<s>[INST] how to solve $|z|^2 + z|z| + 2 = 2i Im(z)$? this is the process I follow:

*

*$z|z|  = 2iImz -|z|^2- 2 $

*$z \sqrt{Re(z)^2 + Im(z)^2} = 2iy - (Re(z)^2 + Im(z)^2) - 2$

*$z\sqrt{x^2 + y^2} =2iy -(x^2+y^2)-2$

*$z=(2iy -(x^2+y^2)-2)/\sqrt{x^2 + y^2}$
then i get: $Im(z)=2y/\sqrt{x^2 + y^2}$ and $Re(z)=(-(x^2+y^2)-2)/\sqrt{x^2 + y^2}$
what should i do next?
 [/INST] Write $z=re^{i\theta}=r(\cos\theta + i \sin\theta)$ with $r,\theta\in\Bbb{R}$; your equation becomes $r^2+r\times r(\cos\theta+i\sin\theta)+2=2ir\sin\theta$.
Identifying real and imaginary parts of the left & right members, you get:
$\begin{cases}
r^2(1+\cos\theta)+2=0&(1)\\[5pt]
r^2\sin\theta=2r\sin\theta&(2)
\end{cases}$
Now (1) implies that there is no solution to the problem… (Or did I make a mistake?)
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