Question

Find all non-negative integral solutions of $y^2+6xy-8x=0.$

Answer 1

solution :

Integral solution of $y^2+6xy-8x=0$

for fixed $x,y=\frac{-6x\pm \sqrt{36x^2+32x}}{2}$

$=-3x\pm \sqrt{9x^2+8x}$

If $x$ is negative, then to make $y$ negative as well

as we would need to choose the branch and

have $9x^2+8x>9x^2$

but that doesn't happen in this case. So no integral negative solutions.