<s>[INST] Polar form of a complex with square root 
Find the polar form of the roots of the polinomial
$p(z)=iz^2-2z+1+2i$

I found the roots $z_1=-i-\sqrt{i-3}$ and $z_2=-i+\sqrt{i-3}$ but I don't know how to deal with the complex numbers inside the square root in order to isolate both real and imaginary parts.
 [/INST] To compute the square root of a complex number: $(x+iy)^2=-3+i$, expand the l.h.s. and identify the real and imaginary parts:
$$x^2-y^2=-3, \quad xy=\frac12.$$
You simplify the  computation observing the square of the modulus of $x+iy$ is the modulus of $-3+i$:
$$x^2+y^2=\sqrt{10}.$$
So we have a linear system in $x^2$ and $y^2$:
$$\begin{cases}x^2+y^2=\sqrt{10}\\x^2-y^2=-3\end{cases}\iff\begin{cases}x^2=\dfrac{\sqrt{10}+3}2\\ y^2=\dfrac{\sqrt{10}-3}2\end{cases}$$
Observe $xy>0$, so $x$ and $y$ have the same sign, and   ultimately
$$x+iy=\pm\frac12\biggl(\sqrt{2\sqrt{10}+6}+i\sqrt{2\sqrt{10}-6}\biggr).$$
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