Bhaskara's Lemma

Lemma
Let $m \in \Z$ be an integer.

For $k \ne 0$:
 * $N x^2 + k = y^2 \implies N \paren {\dfrac {m x + y} k}^2 + \dfrac {m^2 - N} k = \paren {\dfrac {m y + N x} k}^2$

Proof
The implication goes the other way if $m^2 - N \ne 0$.

Also see
This lemma is used when deriving the Chakravala Method of solving indeterminate quadratic equations.