The Discriminant

The discriminant is a key component in the quadratic formula used to determine the nature of the roots of a quadratic equation.

Finding the discriminant

The discriminant \(D\) is given by the formula:

\[D = b^2 - 4ac\]

If this looks familiar, that is because it is the bit under the square root in the quadratic formula.

Interpreting the Discriminant

The value of the discriminant helps to determine the number and type of roots for the quadratic equation:

If \(D > 0\): The equation has two distinct real roots. The graph intersects the x-axis at two points.

If \(D = 0\): The equation has one real root (a repeated root). The graph touches the x-axis at one point.

If \(D < 0\): The equation has no real roots (the roots are complex). The graph does not intersect the x-axis.

This occurs because you cannot square root a negative and get a real number (no solutions). Also, adding 0 and subtracting 0 will give the same answer (one solution).