Rewriting this complex square root for some reason I can to figure out how to rewrite this square root.
I have:
$\sqrt{2+i}$
And I need to rewrite it into:
$\frac{\sqrt{2(\sqrt{5} + 2)} + \sqrt{-2(\sqrt{5} - 2)}}{2}$
Can anybody show me I to do this? I've been trying for an hour now..

One can use trigonometry for sure but I thing the fastest way to solve this is just to solve the quadratic.
$$(x+yi)^2=2+i$$ so 
$$x^2-y^2=2$$
$$2xy=1$$
and $$(x^2+y^2)^2=(x^2-y^2)^2+4x^2y^2=4+1=5$$
So $$x^2+y^2=\sqrt{5}$$
(positive since $x$ and $y$ real)
and 
$$x^2=\frac{\sqrt{5}+2}{2}$$
$$y^2=\frac{\sqrt{5}-2}{2}$$
so 
$$x=\pm\sqrt{\frac{\sqrt{5}+2}{2}}$$
$$y=\pm\sqrt{\frac{\sqrt{5}-2}{2}}$$
Both $x$ and $y$ have same sign since $2xy=1$.
