Solve for $z$ if $z^{2} + | z | = 0$

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Solution

Let $z = x + i y$
$\Rightarrow$ ${(x + i y)}^{2} + \sqrt{x^{2} + y^{2}} = 0$
$\Rightarrow$ $x^{2} - y^{2} + \sqrt{x^{2} + y^{2}} = 0$ and $2 x y = 0$
$\Rightarrow$ $x = 0$ or $y = 0$
when $x = 0$ $- y^{2} + | y | = 0$
$\Rightarrow$ $y = 0, 1, - 1$
$\Rightarrow$ $z = 0, i, - i$
when $y = 0$ $x^{2} + | x | = 0$
$\Rightarrow$ $x = 0$
$\Rightarrow$ $z = 0$
$z = 0, z = i, z = - i$

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