I'm gonna teach you the 'normal' way of doing it, Drongoski showed you the shortcut, I'm building off of Psmao's method.
So, we are given that
z^2=2-\sqrt{5}i
Now remember that the complex number z can be expressed as z=x+iy where x,y\in\mathbb{R},
(x+iy)^2=2-\sqrt5i...