By an integral binary quadratic form (IBQF for short) I mean an $$f(x,y) = ax^2 + bxy + cy^2$$ with $a,b,c \in \mathbb{Z}$. Note that I am not assuming that they are all coprime.
Such an $f$ is said to be reducible if it factors as the product of two linear forms; equivalently, if its discriminant, defined as $\Delta(f) := b^2 - 4ac$, is a square in $\mathbb{Z}$.
I would like to say that, if two IBQFs $f$ and $g$ are reducible and have the same discriminant, then they must be equivalent, by which I mean that there exists a matrix $\gamma \in SL_2(\mathbb{Z})$ such that $$\gamma f = g.$$
Is this true?