How do I compute the eleventh roots of unity algebraically, expressed with only square roots and fifth roots?

Question

I was able to reduce the problem to solving the quintic equation $$x^5+x^4-4x^3-3x^2+3x+1=0,$$ which seems to be the minimal polynomial of $2\cos(2\pi/11)=e^{2\pi i/11}+e^{-2\pi i/11}$. But I couldn't find a method to go further with my calculation. I haven't studied Galois theory, although I know some basic field theory.