Nested Radical of Period $(++-)$

Clearing all the square roots leads to a messy degree $8$ polynomial, which is only solvable numerically (realistically - it does factorise into a quadratic and two cubics, but this is not obvious). It also has eight real roots, seven of which are not the solution we're looking for.

Since a numerical approach is needed anyway, a much easier method is to simply pick a sensible start point $x_0$ , and iterate the given function as $x_{n+1}=\sqrt{4+\sqrt{4+\sqrt{4-x_n}}}$ for $n \ge 1$ . It's not guaranteed this will converge, but in this case we're lucky.

What might a sensible start value be? Well, by inspection, $x$ must be in the interval $(0,4)$ , so $x_0=2$ isn't a bad guess.

Plugging in to the above scheme, we find $x_1=2.51532\ldots,x_2=2.50687\ldots,x_3=2.50702\ldots$ , quickly converging on the solution.