This problem was in my calculus exercise, yet I don't understand why:

Solve the equation :

$\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}$

(Show there are not other solutions to the ones you found)

Now, solving it regularly using simple algebra got me into an eighth-degree polynomial, which reminded my my teachers have just told us there are proofs for why there are no 'formulas' for solving polynomial equations higher than 5.

Anyway, how can I solve this problem using calculus tools? I thought of doing the following thing:

We know that $\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}$, therefore:

$\displaystyle x=\sqrt{5+\sqrt{5+\sqrt{5+x}}}=\sqrt{5+\sqrt{5+\sq rt{5+\sqrt{5+\sqrt{5+\sqrt{5+x}}}}}}$.

Let the first expression be $\displaystyle a_3$, ane the second expression be $\displaystyle a_6$. Therefore:

$\displaystyle x=a_{3n}=\sqrt{5+\sqrt{5+\sqrt{5+...}}}$

Therefore,

$\displaystyle lim(x)=lim(a_{3n})$

but how do I find it ?