\begin{align*} \sigma^2 & = \frac{\sum_{k=1}^n ( x_k-\mu )^2}{n}\\ & = \left ( \frac{\sum_{k=1}^n x_k^2 }{n} \right ) - \mu^2\\ & = \left ( \frac{\sum_{k=1}^n x_k (x_k - 1) + x_k }{n} \right ) - \mu^2\\ & = \left ( \frac{\sum_{k=1}^n x_k (x_k - 1)}{n} \right ) + \mu - \mu^2 \end{align*}