Finding all roots of $x^4-10x^2+1$ without computational approximation

Question

How can I find, by hand, all four roots of $f(x)=x^4-10x^2+1$?

I can see that it has no linear rational roots, by the root test, and I can't see anyway to factorize it into two quadratic expressions. I would guess that it is irreducible in $\Bbb Q[x]$, so I just want to see what four roots it has in $\Bbb R$ or $\Bbb C$ without using computational methods.

Side note: (I know that it splits over $\Bbb Q(\sqrt{2},\sqrt{3})$, so I wanted to see how hard it is to find all four roots, and drew a blank. If I showed all four roots of $f(x)$, which is the minimal polynomial of $\sqrt{2}+\sqrt{3}$, were in $\Bbb Q(\sqrt{2},\sqrt{3})$ I imagine I would have shown $\Bbb Q(\sqrt{2}+\sqrt{3})=\Bbb Q(\sqrt{2},\sqrt{3})$)

Answer 1

Hint:

Replace $x^2$ with some variable $y$. You now have a quadratic equation that you can solve using the quadratic formula.

Answer 2

$x^4-10x^2+1=0\iff (x^2-5)^2=24\iff x^2=5\pm2\sqrt 6\iff $ $x\in \{\sqrt 3-\sqrt 2,\sqrt 3+\sqrt 2,-\sqrt 3-\sqrt 2,-\sqrt 3-\sqrt 2\}$.