<s>[INST] If $y =\sqrt{5+\sqrt{5-\sqrt{5+ \cdots}}}$, what is the value of $y^2-y$? If $$y =\sqrt{5+\sqrt{5-\sqrt{5 + \cdots}}},$$
what is the value of $y^2-y$ ?
I am unable to get the clue due to these alternative signs of plus and minus please help on this thanks...
 [/INST] Hint: 
$$\begin{align}
(y^2 -5)^2 - 5 &= -y \\
y^4 - 10y^2 + 25 - 5 + y &= 0 \\
y^4 -10y^2 + y + 20 &= 0.
\end{align}
$$
Edit: To elaborate a bit on the comments given below: you have produced a depressed quartic equation. As explained on the Wikipedia article you can try to factor this into quadratics
$$
y^4 -10y^2 + y + 20 = (y^2 +py + q)(y^2 + ry + s).
$$
All you have to do is find $p,q,r$, ands $s$. You get
$$\begin{align}
y^4 -10y^2 + y + 20 &= (y^2 +py + q)(y^2 + ry + s)\\
&=  y^4 + (p + r)y^3 + (q + s + pr)y^2 + (ps + qr)y + qs.
\end{align}
$$
Now put coefficients equal to each other and solve ...
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