Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

Question

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution.

We know that the first moment is equal to:

$\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$

and the second moment is equal to:

$\mu_2 = \frac{\alpha_1(\alpha_1 +1)}{(\alpha_1+\alpha_2)(\alpha_1+\alpha_2+1)}$

The solutions for $\alpha_1$ and $\alpha_2$ are:

$\alpha_1 = \frac{\mu_1-\mu_2}{\mu_2-1}$ , $\hspace{10mm}\mu_2 = \frac{\mu_1-\mu_2}{\mu_2-1}\frac{1-\mu_1}{\mu_1}$

I have attempted the algebra several times and I am quite close to the solution, but unfortunately I'm still off.

Answer 1

Try with the correct $\mu_2$, which is $$\mu_2 = \frac{\alpha_1(\alpha_1 +1)}{(\alpha_1+\alpha_2)(\alpha_1+\alpha_2+1)}.$$