Question

Number of solution's of the equation $z^2+|z|=0,$ where $z\in C$ is

Answer 1

Let $z=x+iy$
$z^2+|z|=0$
$\Rightarrow x^2-y^2+2xyi+\sqrt{x^2+y^2}=0$
comparing real and imaginary parts
$\Rightarrow x^2-y^2+\sqrt{x^2+y^2}=0$
and $2xy=0\Rightarrow x=0$ or $y=0$

$\text{Case}-1:$ When $x=0$
$\Rightarrow -y^2+\sqrt{y^2}=0$
$\Rightarrow y^2=|y|$
$\Rightarrow y^2=\pm y$
$\Rightarrow y=0,\pm 1$
$\therefore$ solutions are $(0,0)(0,1),(0,-1)$

$\text{Case}-2:$ When $y=0$
$\Rightarrow x^2+\sqrt{x^2}=0\Rightarrow x=0$
$\therefore$ Solution is $(0,0)$
Hence, solutions of the equations are $(0,0),(0,1),(0,-1)$