Discriminant of a Quadratic

What is the discriminant?

The discriminant of a quadratic is closely linked to the Quadratic Formula. For a general quadratic \(ax^2+bx+c\), the discriminant, \(D\), is \[D=b^2-4ac\] You can easily see where this comes from in the Quadratic Formula \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

\(D>0\)
Two distinct roots
\(D=0\)
One repeated root
\(D<0\)
No real roots

The case \(D<0\) can be seen to have no real roots because in the quadratic formula, it would result in square rooting a negative value which cannot give a real answer.

Visual examples for various discriminants

\(D>0\)
Graph with no real roots
Graph with no real roots
\(D=0\)
Graph with repeated root
Graph with repeated root
\(D<0\)
Graph with distinct roots
Graph with distinct roots