solve:
cos^2(x) + cos(x) = cos2x
If possible; Otherwise, resort to quadratic formula.
Also, remember this is a quadratic in terms of $\displaystyle \cos x$, so once you have your solutions, you need to solve $\displaystyle \cos x=u_1$ and $\displaystyle \cos x=u_2$ to find the solutions that you're seeking.
$\displaystyle u^2 - u - 1 = 0$
$\displaystyle u^2 - u + \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right)^2 - 1 = 0$
$\displaystyle \left(u - \frac{1}{2}\right)^2 - \frac{5}{4} = 0$
$\displaystyle \left(u - \frac{1}{2}\right)^2 = \frac{5}{4}$
$\displaystyle u - \frac{1}{2} = \frac{\pm \sqrt{5}}{2}$
$\displaystyle u = \frac{1 \pm \sqrt{5}}{2}$.
Note that $\displaystyle -1 \leq \cos{x} \leq 1$ for all $\displaystyle x$. Are there any solutions that don't fit?