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 $2xy=1$.