I know how to solve linear diophantine equations, but I was wondering if someone can give me a step by step to solve something like $2x^2 + 2x - 5y = -1$? I cannot find a lot of resources on this particular form.
I know the solutions are
$${ y = 10 k^2 - 14 k + 5, x = 3 - 5 k}$$
$${ y = 10 k^2 - 6 k + 1, x = 1 - 5 k}$$
$$ 2 x^2 + 2 x + 1 \equiv 0 \pmod 5 $$ $$ 4 x^2 + 4 x + 2 \equiv 0 \pmod 5 $$ $$ (4 x^2 + 4 x + 1) +1 \equiv 0 \pmod 5 $$ $$ (4 x^2 + 4 x + 1) \equiv -1 \pmod 5 $$ $$ (2 x + 1)^2 \equiv -1 \pmod 5 $$ $$ 2 x + 1 \equiv 2,3 \pmod 5 $$ $$ 2 x \equiv 1,2 \pmod 5 $$ $$ 2 x \equiv 6,2 \pmod 5 $$ $$ x \equiv 3,1 \pmod 5 $$