Bhaskara's lemma

${\displaystyle N{x}^2+k=y^2⟹N{\left(\frac{mx+y}{k}\right)}^2+\frac{m^2-N}{k}={\left(\frac{my+Nx}{k}\right)}^2}$

for integers ${\displaystyle m,x,y,N,}$ and non-zero integer ${\displaystyle k}$.

Proof

The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by ${\displaystyle m^2-N}$, add ${\displaystyle N^2{x}^2+2Nmxy+N{y}^2}$, factor, and divide by ${\displaystyle k^2}$.

${\displaystyle N{x}^2+k=y^2⟹N{m}^2{x}^2-N^2{x}^2+k(m^2-N)=m^2{y}^2-N{y}^2}$

${\displaystyle ⟹N{m}^2{x}^2+2Nmxy+N{y}^2+k(m^2-N)=m^2{y}^2+2Nmxy+N^2{x}^2}$

${\displaystyle ⟹N(mx+y{)}^2+k(m^2-N)=(my+Nx{)}^2}$

${\displaystyle ⟹N{\left(\frac{mx+y}{k}\right)}^2+\frac{m^2-N}{k}={\left(\frac{my+Nx}{k}\right)}^2.}$

So long as neither ${\displaystyle k}$ nor ${\displaystyle m^2-N}$ are zero, the implication goes in both directions. (The lemma holds for real or complex numbers as well as integers.)