The Variance Shortcut Formula

By definition, $\mathrm{Var}(X) = E[(X-\mu)^2]$ where $\mu = E[X]$. Expand the square inside the expectation: $(X-\mu)^2 = X^2 - 2\mu X + \mu^2$.

Apply linearity of expectation term by term: $E[(X-\mu)^2] = E[X^2] - 2\mu E[X] + \mu^2$. Since $\mu$ is a constant (it does not depend on the outcome), it can be pulled out of the expectation in the middle and last terms.

Now substitute $E[X] = \mu$ back in: $E[X^2] - 2\mu\cdot\mu + \mu^2 = E[X^2] - 2\mu^2 + \mu^2 = E[X^2] - \mu^2$. Replacing $\mu$ with $E[X]$ gives exactly $\mathrm{Var}(X) = E[X^2] - E[X]^2$, as claimed.