Solve an equation with a finite chain of nested radicals

Question

Solving an infinitely long square root problem is easy but how to solve this one? The equation goes like this. $$\sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4-x}}}} = x$$

Answer 1

Its around 2.526062179 and Wolfram Alpha agrees.

You can come to the same result by visiting "Wolfram Alpha". You can also rearrange the term stepwise:

$$ x=\sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4-x}}}} $$ $$ x^2-4=\sqrt{4+\sqrt{4-\sqrt{4-x}}} $$ $$ (x^2-4)^2-4=\sqrt{4-\sqrt{4-x}} $$ $$ 4-(4-((x^2-4)^2-4)^2)^2=x $$ When you now resolve the parentheses you get: $$ \small{x^{12}-24 x^{10}-x^9+228 x^8+16 x^7-1087 x^6-88 x^5+2720 x^4+191 x^3-3380 x^2-136 x+1633} = 0 $$ Which Wolfram Alpha resolves too: $$ x \approx 2.5260621791756334159 $$