Proving existence of roots

Question

I have the following arbitrary function which is the result of solving an iterative map for any period two fixed points (ie. for $g(x_n) = x_{n+1}$, I am trying to find $k$-values for which g(g(x)) = x) where $T,B,k \in \mathbb{R}_{> 0}$: $$T^k x^{k^2-k} - (B-x^k)^{k-1} - T(T-1)(B-x^k)^{k-2}x^k = 0$$ and I am trying to identify the $k$ values for which 2-period bifurcation occurs. In simulation, we have a stable fixed point with dampened oscillations to this point for $k < 2$, and period doubling followed by chaos as $k$ increases past 2. We additionally always have a fixed point at $$x = \left(\frac{B}{T+1}\right)^{1/k}$$ which is stable for $0 < k < 2$. How should I proceed with finding proving this bifurcation?