Question

Find all complex numbers $z$ which satisfy the following equation
$z^2+\left|z\right|=0$.

Answer 1

$x^2-y^2+i2xy+\sqrt{x^2+y^2}=0$

$\therefore x^2-y^2+\sqrt{x^2+y^2}=0$

$2xy=0$ $\therefore x=0$ or $y=0$
When $x=0$ then $-y^2\pm y=0$ .. $y=0,\pm 1$
$\therefore (0,0),(0,1),(0,-1)$ or $z=0,i,-i$
When $y=0$ then we will get $(0,0)$, $(1,0)$ and $(-1,0)$.
But these not complex numbers.
Hence $0,i,-i$ are the only three solutions of the given equation.