Condition on a quadratic equation to be an odd perfect square using modular arithmetic

Question

As a result of the discussion in this question Quadratic residues and squares of odd numbers, @Mark Bennet asked me to open this question. I have a quadratic expression $14144x^2+3872x+265$. How can I prove that this is never a perfect square using modular arithmetic or otherwise? In general how can any expression $ax^2+bx+c$ be checked if it is an odd perfect square or not?

Answer 1

To solve

$$ax^2+bx+c=y^2$$

we first multiply both sides by $4a$ to obtain

$$4a^2x^2+4abx+4ac=4ay^2$$

In other words,

$$(2ax+b)^2-(4a)y^2=(b^2-4ac)$$

This is a generalised Pell's equation, which is very nontrivial to solve, e.g http://math.stanford.edu/~conrad/154Page/handouts/genpell.pdf

Because this generalised Pell's equation is equivalent to the original, there's no better way to solve the equation for general $a,b,c$. Of course, in specific instances it is easy to prove that no solutions exist (mods, bounding or otherwise).