Let $\displaystyle \alpha = z+z^{-1}. $

show that $\displaystyle \alpha^{2} + \alpha -1 = 0 $

I have tried to square the first equation and get,

$\displaystyle \alpha^{2} = z^{2} + z^{-2} +2 $

and i can't seem to go any further from here, any help appreciated.