Discrepancy in Definition of Discriminant of Binary Quadratic Form

Question

Given a quadratic form, the discriminant is usually defined as the determinant of the associated symmetric matrix. However, I've seen a number of authors define the discriminant associated to the binary quadratic form $ax^2 + bxy + cy^2$ to be $4ac - b^2$, when the determinant of the associated symmetric matrix is $ac - b^2/4$. Is there any good reason to take one definition of discriminant over the other?