If discriminant of f is a perfect square, then we can factor f into linear factors

Question

Let $f$ be a binary quadratic form with integer coeficients, $f(x,y)=ax^2+bxy+cy^2$. I'm trying to prove that if $d=b^2-4ac$ is a perfect square $d=k^2$, then we can factor $f(x,y)=(a_1x+a_2y)(a_3x+a_4y)$ with $a_i \in \mathbb{Z}$.

I tried various manipulations, and I find something that is almost what I want, mainly $4af(x,y)=(2axy+by)^2-dy^2=(2axy+by-ky)(2axy+by+ky)$, but I didn't get much further...

Thank you!

Answer 1

Hint: Let $z = \frac xy$. We can write $$ f(x,y) = y^2(az^2 + bz + c) $$ Using the quadratic formula, how can we factor $az^2 + bz + c$, assuming the discriminant is a perfect square?