For the following equation,
$$ \sum_{x=b}^{N}(1-\alpha)^{x-1}\alpha^{N-x} \binom{N-1}{x-1} \left( (N+1) \alpha + x-1-N \right) = 0, $$
where $b \in \mathbb{N}$ with $b \leq N$. Simulation shows that this equation has a unique $\alpha^* \in [0, 1)$. $1$ is not included since $\alpha^* = 1$ always satisfies the equation. Do you guys have any sense how to show it is the case?