Equations with Radicals Let $a$, $b$, $c$, and $d$ be distinct real numbers such that
\begin{align*}
a &= \sqrt{4 + \sqrt{5 + a}}, \\
b &= \sqrt{4 - \sqrt{5 + b}}, \\
c &= \sqrt{4 + \sqrt{5 - c}}, \\
d &= \sqrt{4 - \sqrt{5 - d}}.
\end{align*}
Compute $abcd$.
I squared it to 
$a^4-8a^2-a+11=0,$
$b^4-8b^2-b+11=0,$
$c^4-8c^2+c+11=0,$
$d^4-8d^2+d+11=0,$
But I'm stuck here.
Could I get a full answer instead of a hint?  Thanks!

By your calculations, $a, b, -c, -d$ are exactly the roots of $x^4 -8x^2 - x + 11$. (We know they're distinct because $a, b, c, d$ are distinct and positive.) So we can use these formulas to find that the product of the roots of this polynomial is $11$ and hence $abcd = ab(-c)(-d) = 11$.
