What is the extinction process of a branching process with $Bin(2, 1-p)$ offspring distribution?
My attempt:
$G_n(s) = E(s^{Z_n})$ is the PGF of $Z_n$, the population size at time $n$
Let $G(s) = E(s^Y)$, $Y~Bin(2, 1-p)$, so the PGF is $G(s) = (ps+q)^n$
Want to solve $G(s)=s$
$G(s) = ((1-p)s+p)^2 = s^2-2s^2p+2sp+s^2p^2s-2sp^2+p^2$
Now I'm having trouble solving:
$s^2-2s^2p+2sp+s^2p^2s-2sp^2+p^2 = s$