Hungerfords Algebra Definition 4.4:
Let $K$ be a field with char$K \neq 2$ and $f \in K[x]$ be a polynomial of degree $n$ with $n$ distinct roots $u_1,..u_n$ in some splitting field $F$ of $f$ over $K$. Let $\Delta=\prod_{i<j}(u_i-u_j)$; the discriminant of $f$ is $D=\Delta^2$.
I learned that the definition of a quadratic $g(x)=ax^2+bx+c$ is $D=b^2-4ac$. Hungerfords definition of a discrimant does not lead to this discrimiant. The roots of $g(x)$ are $u_i=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
However:
$(u_1-u_2)^2 = \frac{b^2-4ac}{a^2}$.
Is this just an unfortunate inconsistency in mathematical terminology? Discriminant's are something that I've wanted to know more about for some time now so that would be a little disappointing to me