+118608 Let $z$ be a complex number such that z^2 + |z|^2 = 3 - 5i.Find |z|^2.
\(Let \;\;z=a+bi \\ z^2= a^2-b^2+2abi\\ |z|^2=a^2+b^2\\ so a^2-b^2+2abi + a^2+b^2 = 3-5i \\ 2a^2+2abi = 3-5i \\ 2a^2=3 \qquad 2ab=-5\\ a^2=1.5 \qquad 2a^2b^2=25\\ \qquad \qquad \quad 2*1.5b^2=25\\ \qquad \qquad \quad b^2=\frac{25}{3}\\ so\\ |z|^2=\frac{3}{2}+\frac{25}{3}=\frac{106}{6}=17\frac{2}{3}\)
As always, you must check this answer.
