Help Solving a Quartic Polynomial

[DuncanM]

I am trying to solve an equation:

arccos(Y) = arctan(Y), where Y = 1/x

This turns into a quartic equation:

x⁴ - x² - 1 = 0

It looks simple enough to simplify further; however, I must be having a brain-fart because I can't do it. I'd like to avoid resorting to a numerical solution.

Am I overlooking something?
Can this equation be simplified further? If so, how?

Any help is much appreciated.

 

[Number Nine]

Setting $u = x^2$, you can rewrite it as $u^2 - u - 1 = 0$. You can solve this easily enough for the roots in u, and then solve the roots for x.

 

[DuncanM]

Excellent!

Thank-you very much!