Sqrt(x)=-x

x=0

Right? It can't be positive or negative so it has to be 0.

looks good.

You could arrange as:

$x+\sqrt{x}=0$

$\sqrt{x}(\sqrt{x}+1)=0$

Since $0\le\sqrt{x}$ we must have:

$\sqrt{x}=0$

$x=0$