Question

Solve the equation $z^2+|z|=0$, where $z$ is a complex number.

Answer 1

Simplifying $z^2+\left|z\right|=0$

$z^2=-\left|z\right|$

$\left|z^2\right|=\left|z\right|$

${\left|z\right|}^2-\left|z\right|=0$

$\left|z\right|\left(\left|z\right|-1\right)=0$

$\left|z\right|=0,1$

If $\left|z\right|=0$, then $z=0$ and if $\left|z\right|=1$, then,

$z^2=-1$

$z=\pm i$

Therefore, the solutions are $0,i,-i$.