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?


$$a+bi+\sqrt{a^2+b^2} = 2 + 8i$$
  $$a^2-b^2+a^2+b^2=4-64$$

This is where you went wrong, it looks like you tried to square both sides and take the real part but you made the wrong assumption that all the cross-multiplication terms would be imaginary.
$$a+bi+\sqrt{a^2+b^2} = 2 + 8i$$
$$a^2-b^2+(a^2+b^2) + 2abi + 2a\sqrt{a^2+b^2} + 2b\sqrt{a^2+b^2}i = 4 - 64 +16i$$
$$a^2-b^2+a^2+b^2 + 2a\sqrt{a^2+b^2}  = 4 - 64 $$
$$2ab + 2b\sqrt{a^2+b^2} = 16$$
Which frankly doesn't look much like progress towards a soloution. We have a pair of equations but both of them still contain our problematic square root.
To eliminate the square root we need to move it to it's own side of the equation before squaring.
$$a+bi+\sqrt{a^2+b^2} = 2 + 8i$$
$$\sqrt{a^2+b^2} = (2 - a)+ (8- b)i$$
$$a^2+b^2 =(2 - a)^2 - (8- b)^2 + 2(2 - a)(8- b)i $$
$$a^2+b^2 = a^2 - 4a + 4 - b^2 +8b -64 $$
$$2b^2 =- 4a + 4  +8b -60 $$
$$0 = (2 - a)(8- b)$$
