Solve equation with absolute value of complex numbers Find all complex numbers $z$ such that $|z| + \overline{z}=9-3i$.
Thinking about using with the fact that $z=a+bi$ and $|z| = \sqrt{a^2+b^2}$ in some way. Not sure how.

Write $\;z=x+iy\;$ , so that your equation is
$$\sqrt{x^2+y^2}+x-iy=9-3i\stackrel{\text{compare real and imag. parts}}\implies\begin{cases}\sqrt{x^2+y^2}+x=9\\{}\\y=3\end{cases}\;\;\implies$$
$$\sqrt{x^2+3^2}+x=9\implies x^2+9=81-18x+x^2\implies18x=72\implies x=4$$
and the unique solution is $\;4+3i\;$
