Variance of Binomial Distribution/Proof 1

Proof
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $\var X = \expect {X^2} - \paren {\expect X}^2$

From Expectation of Function of Discrete Random Variable:
 * $\ds \expect {X^2} = \sum_{x \mathop \in \Img X} x^2 \Pr \paren {X = x}$

To simplify the algebra a bit, let $q = 1 - p$, so $p + q = 1$.

So:

Then:

as required.