<s>[INST] $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the nearest value of $y^2 - y$? I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question.


$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $,  what is the
    nearest value of $y^2 - y$?
$\textbf {(1) } 1 \qquad \textbf {(2) } \sqrt5 \qquad \textbf {(3) } 4 \qquad \textbf {(4) } 2\sqrt5$


I worked the problem and got $$y^2 - y = 5+ \sqrt{5-\sqrt{5+\sqrt{5-...}}} -\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} \\ = 5+ \sqrt{5-y} - y = 5-y + \sqrt{5-y}$$
What do I do next?
 [/INST] You were on the right track. Put
$$x=\sqrt{5+\sqrt{5-\sqrt{5+\ldots}}}\implies x^2=5+\sqrt{5-\sqrt{5+\ldots}}\implies$$
$$x^4-10x^2+25=5-\sqrt{5+\sqrt{5-\ldots}}\implies$$
$$x^4-10x^2+20=-\sqrt{5+\sqrt{5-\ldots}}=-x\implies\;\ldots\ldots$$
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