Why is the discriminant of a quadratic what it is? I of course want to apply this to an nth degree polynomial.

Question

Why do I keep getting that the discriminant of a quadratic is $4a^2c - ab^2$ when I use sylvester matrices? I know that I can factor out -a and get $-a(b^2-4ac)$. But this is going to be negative when the actual discriminant is positive and positive when it should be negative. In fact, this only agrees with the actual one when the two roots are equal. I need to expand this to higher degrees so it'd be nice to know what's wrong here.