Question

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

Answer 1

Let $z=x+iy$

Then, $z^2+|z|=0$

$⟹(x+iy{)}^2+\sqrt{x^2+y^2}=0$

$⟹x^2-y^2+2xyi+\sqrt{x^2+y^2}=0+0i$

By comparing real an imaginary parts:

$x^2-y^2+\sqrt{x^2+y^2}=0$..........[1] or

$2xy=0⟹x=0$ or $y=0$

Case$1$: if $y=0$, from eq. $1$

$x^2+\sqrt{x^2}=0⟹x^2+|x|=0$..........[2]

Now if $x>0$, then $x^2+x$ will not be $0$(sum of two positive numbers can't be $0$).

So, for $x<0$; $x^2-x=0⟹x=0,1$. But $x=1$ doesn't satisfy eq. 2.

So only point that satisfies is $y=0,x=0$

Case$2$: if $x=0$, from eq. $1$

$-y^2+\sqrt{y^2}=0⟹-y^2+|y|=0$..........[3]

Now if $y<0$, then $-y^2-y=0⟹y=0,-1$

So, for $y>0$; $-y^2+y=0⟹y=0,1$. But $y=1,0$ both satisfies eq. 3.

So the points that satisfy case$2$ are $y=0,x=0$ and $x=0,y=1$ and $x=0,y=-1$

Hence all points which satisfies the given equation $z^2+|z|=0$ is $(0,0),(0,1)$ and $(0,-1)$

So,

$z=0+0i=0$

$z=0+1i=i$

$z=0-1i=-i$