I have the PGF: $$ G_x(\theta) = \frac{a-1}{a-\theta^2}\;\; a>1 = \text{constant} $$
I am trying to find the ultimate extinction probability $e$
and I think I need to solve $$e=G_x(e).$$ however I am up to the line $$e^3 - ae + (a-1) =0,$$ and I am unsure as to how I may proceed
Could you please help I am trying to work out if the fish in my fish tank will go extinct or not
matter of urgency
thanks