<s>[INST] Repeating Square Root Closed Form I've been thinking about repeating square roots: 
$\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$. I wrote a program on my calculator to do it $n$ times and I found that, if $x = y^2 - y$ then $\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$ approaches $y$ as $n$ gets large. So, solving for $y$:
$$y^2 - y - x = 0$$
$$y =  \frac{1 \pm \sqrt{1 + 4x}}{2} = \frac{\sqrt{4x + 1} + 1}{2}$$
$$\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}} =\frac{\sqrt{4x + 1} + 1}{2} $$
This works, but where does the $y^2-y=x$ come from? I got here via plugging and guessing, but how can I prove that with an input of $x$ where $x = y^2 - y$, the answer will be $y$?
 [/INST] Define
$$\alpha:=\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}=\sqrt{x+\alpha}\implies \alpha^2-\alpha-x=0\ldots$$
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