Let $(X_n)_{n\geq0}$ be a branching process, where $X_0=1$
Given $\mathbb{P}(Z=z)= \begin{cases} a & z=0 \\ b & z=1\\ c &z=2 \end{cases}$, where $a,b,c \in (0,1)$ , $a+b+c=1$ and $c<a$
My Attempt:
$G(S) = \mathbb{E}(s^Z) = \sum_{s=0}^\infty s^Z\mathbb{P}(Z=z)=s $
$=a+bs+cs^2=s$
$=a+(b-1)s+cs^2=0$
Then, I know that $s=1$ is a solution. How do I find a,b, and c for the other solution / an equation for them?
Or have I misunderstood what to do?