Find the value of this infinitely nested radical (it appears to obtain multiple values) 
Find the value of $$\sqrt{1-\sqrt{\frac{17}{16}-\sqrt{1-\sqrt{\frac{17}{16}-\cdots}}}}$$

This is not as simple as it looks for one reason - there are $2$ real solutions to the equation $$x=\sqrt{1-\sqrt{\frac{17}{16}-x}}\implies\begin{cases}x_1=0.5\\x_2\approx 0.073\end{cases}$$
You can see that for yourself on Wolfram Alpha.
How can we know the real value of this infinite radical?

Solution technique:
Let 
$$L = \sqrt{1 - \sqrt{\frac{17}{6} - \sqrt{1 - \sqrt{\frac{17}{6} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }} } $$
The natural method of solution is to observe that
$$L^2 = 1 - \sqrt{\frac{17}{6} - \sqrt{1 - \sqrt{\frac{17}{6} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }}$$
So...
$$ L^2  - 1= -\sqrt{\frac{17}{16} - \sqrt{1 - \sqrt{\frac{17}{16} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }}$$
Therefore
$$ (L^2 - 1)^2 = \frac{17}{16} - \sqrt{1 - \sqrt{\frac{17}{16} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }$$
Implying:
$$(L^2 - 1)^2 - \frac{17}{16} = -\sqrt{1 - \sqrt{\frac{17}{16} -\sqrt{1 - \sqrt{\frac{17}{16} -... } } } }$$
And therefore we substitute
$$(L^2 - 1)^2 - \frac{17}{16} = -L$$
Now this equation has 4 possible solutions each expression for $L$ behaves algebraically the same as this nested radical.
The reason is because if you recall from algebra there are two possible $\sqrt{x}$ for any number (a positive and negative root). Now we need to de-nest 2 roots before we can exploit our substitution so what ends up happening is that that are really four possible infinite-radicals (depending on the types of roots we alernated with) that could've yielded this $L$. Your job is to then approximate YOUR problem to a good amount and see which of the 4 solutions to  
$$(L^2 - 1)^2 - \frac{17}{16} = -L$$
Corresponds and even then, we may have more than one converging value if our nested radical trends in a way analogous to
$$1 - 1 + 1 - 1 + 1 - 1...$$
Whose sum can be seen as 1 or 0 depending on how one chooses to enumerate its values.
