Adding Absolute value to a complex number: $ z+| z|=2+8i$ I would like to know my error in this problem.
Find the complex number such that:
$$ z+|z|=2+8i$$
So far, I have:
$$
\begin{split}
a+bi+\sqrt{a^2+b^2} &= 2 + 8i\\
a^2-b^2+a^2+b^2&=4-64\\
2a^2 -b^2 + b^2&=-60\\
a^2&=-30
\end{split}
$$
But I should end up with
$$a^2=-15$$
No matter how hard I try, I can't seem to find what I did wrong. Any suggestions?

As you square and take the real part, it should be 
$$a^2-b^2+a^2+b^2+2a\sqrt{a^2+b^2}=4-64$$
