calculus - Find limit of recursive sequence $a_{n+1} =sqrt{2+ sqrt{a_n} }.$

Let $a_n$ be the sequence in $\mathbb{R}$ given by:

$$ a_1 = \sqrt{2}, \; \; a_{n+1} =\sqrt{2+ \sqrt{a_n} }. $$

Show that this sequence converges and find the limit.

I have already proven that this sequence is increasing and bounded by $2$, so the limit exists.
Now, when I'm trying to find the limit, say $L$, I get that $L$ satisfies the equation:
$$L^2 -2 - \sqrt{L}=0, $$ but I don't actually know how to solve this equation, so I would like to know if there is an easier way to find the limit, and in case I must solve the last equation, how can I solve it?

Thanks in advance!