Estimate variance of AR(2) using Yule-Walker method

Question

We have a AR(2) process $$X_t-\mu=\phi_1(X_{t-1}-\mu)+\phi_2(X_{t-2}-\mu)+\epsilon_t,$$ where $\epsilon_t$ is a white noice process, and I got $$\hat{\phi_1} = \hat{\rho}(1)\frac{1-\hat{\rho}(2)}{1-\hat{\rho}^2(1)}$$ and $$\hat{\phi_2} = 1-\frac{1-\hat{\rho}(2)}{1-\hat{\rho}^2(1)}.$$ Now I am struggling to get the following $$\hat{\gamma}(0)=\sigma_\epsilon^2\frac{1-\hat{\phi}_2}{(1+\hat{\phi}_2)[(1-\hat{\phi}_2)^2-\hat{\phi}_1^2]}$$ where $\sigma_\epsilon^2$ denote $\operatorname{Var}{(\epsilon_t)}$. $$\implies\sigma_\epsilon^2 = \hat{\gamma}(0)\frac{(1+\hat{\phi}_2)[(1-\hat{\phi}_2)^2-\hat{\phi}_1^2]}{1-\hat{\phi}_2}$$