Question

Indicate the point of the complex plane $z$ which satisfy the following equation.
$z^2+|\overline{z}|=0$

Answer 1

Let $z=x+iy$

$|z|=\sqrt{x^2+y^2}$

$|z|+z^2=0$

$\left(\sqrt{x^2+y^2}\right)$$+x^2-y^2+2ixy=0$

$⟹\left(\sqrt{x^2+y^2}\right)$$+x^2-y^2=0$(Real part) and $2xy=0$(imaginary part)

$⟹(x^2-y^2{)}^2=x^2+y^2$

$⟹x^4+y^4-2x^2{y}^2=x^2+y^2$

$⟹x^4-x^2=y^2-y^4$ (since $xy=0$)

$⟹x^2(-1+x^2)=y^2(1-y^2)$