help in solving $z^2+|z|^2=1+2i$ I can't solve this:
$$z^2+|z|^2=1+2i$$
I've found $\sqrt{ \frac{2}{5}}$ for the modulus of z but I can't explicit about the angle of z . 
The result of the book is, $ \pm (\frac{1}{\sqrt{2}}+i*\sqrt{2})$ but there are so many mistakes that I'm afraid is not very reliable 

$$z^2+|z|^2=(a+bi)^2+a^2+b^2=a^2+2abi-b^2+a^2+b^2=2a^2+2abi$$
So $$2a^2+2abi=1+2i$$
Equating real and imaginary parts:
$$\Re:2a^2=1\Rightarrow a=\pm\frac{1}{\sqrt{2}}$$
$$\Im:2(\pm\frac{1}{\sqrt{2}})bi=2i\Rightarrow\pm\frac{1}{\sqrt{2}}b=1\Rightarrow b=\pm\sqrt{2}$$
Thus,$z=\pm\frac{1}{\sqrt{2}}\pm\sqrt{2}i$
