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...

Consider $z_{-+} = -y = -\sqrt {5 + \sqrt{5 - \ldots}}$.
$z_{-+}$ is clearly a solution to the equation $(z^2 -5)^2-5 = z$,
and the other three solutions of this degree $4$ polynomial equation, should be
$z_{+-} = +\sqrt {5 - \sqrt{5 + \ldots}}, z_{++} = +\sqrt {5 + \sqrt{5 + \ldots}}$, and $z_{--} = -\sqrt {5 - \sqrt{5 - \ldots}}$ .
But $z_{++}$ and $z_{--}$ are in fact solution to a simpler polynomial equation, which is $z^2-5 = z$.
So $z^2-z-5$ has to be a factor of $(z^2-5)^2-z-5$, and doing the division, you end up with $(z^2-5)^2-z-5 = z^4-10z^2-z+20 = (z^2-z-5)(z^2+z-4)$, hence $z_{-+}^2 + z_{-+} - 4 = 0$.
From there you conclude that $y^2-y = z_{-+}^2 + z_{-+} = 4$.
