<s>[INST] Where did the negative answer come from? The question is to evaluate $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$
$$x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$
$$x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots }}}}$$
$$x^2=2+x$$
$$x^2-x-2=0$$
$$(x-2)(x+1)=0$$
$$x=2,-1$$
because $x$ is positive $x=2$ is the answer. but where did the $x=-1$ come from ?
 [/INST] By squaring both sides, you turn an equation with one solution into an equation with two solutions; you can end up creating an extraneous solution. In this case, you come to find that $-1$ satisfies $x^2 = 2 + x$, but does not satisfy $x = \sqrt{2 + \sqrt{2 + \cdots}}$, and so is extraneous.
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