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 $abcd = ab(-c)(-d) = 11$.