I’m looking for a way to find only integer solution pairs to a dual-variable quadratic equation without trial and error.
For example: $$(a+3\sqrt 5)^2+a-b\sqrt 5=51$$
Valid solution pairs are any points on the graph of $(a+3\sqrt 5))^2+a-b\sqrt 5=51$, but the only integer pairs are $(2,12)$ and $(-3,-18)$ Since the above equation is solvable by rearrangement and then comparison(b=6a,a²+a=6), it was suggested that I change my example: a^2+ab^2+2b^2-ab=50 How would one find only integer solutions for this equation(a,b) without trial and error?