Equating different forms of Hellinger Distance For a research report, I want to show that the standard expression of the Hellinger Distance between two discrete distributions,
$$H(p,q)={\sqrt{\frac{1}{2} \sum_{x \in X} \left[\sqrt{p(x)}-\sqrt{q(x)} \right]^{2}}}$$
is equivalent to the alternative expression
$$H(p,q)={\sqrt{1-\sum_{x \in X} \sqrt{p(x)q(x)}}}$$
I have tried the expand the square but can't seem to equate the two expressions. Can anyone help?

Hint: what would $\sum_{x\in \mathcal X} p(x) $ be equal to?
