Complex: $z^2 = 1+2i$ (find real- and imaginary part of $z$) these are my toughs:
$$z^2 = 1 + 2i \Longrightarrow (x+yi)(x+yi) = 1 + 2i$$
so: $x^2-y^2 = 1$ and $2xy = 2$
then i got that $x = 1/y$ but i cant continue to find the real- and imaginary part of z anymore. Appriciated any help

\begin{align}
z^2 = 1 + 2i & = |1+2i|(\cos\alpha+i\sin\alpha) \text{ where } \tan \alpha = \frac 2 1 \\[10pt]
& = \sqrt{1^2+2^2} (\cos\alpha+i\sin\alpha)  = \sqrt 5 (\cos\alpha+i\sin\alpha)
\end{align}
Therefore
$$
z = \pm\left( \sqrt{\sqrt 5} \right)\left( \cos\frac\alpha2 + i \sin\frac\alpha 2 \right).
$$
