How do I solve the Diophantine equation $ax^2 + bx + c = y^2$?
The approach I have so far is to use the transformation $X = 2ax + b$ and $Y = 2y$. Applying this, we get, $X^2 - dY^2 = n$, where $n = b^2 - 4ac$ and $d = a$.
$X^2 - dY^2 = n$ is a Pell equation.
Questions:
- Is there any other method?
- What is the complexity of the algorithm for finding the solution to the Pell equation?