# The Discriminant

## What are roots and the discriminant?

The **discriminant **of a quadratic equation will tell you how many roots the quadratic equation has. For example, in physics, while analyzing projectile motion, one can use the discriminant to predict the number of times the projectile will hit the ground – essentially the roots of the equation. This principal notion is essential not only in quadratic equations but also in higher-order polynomials. Solutions of a quadratic equation are known as **roots**; they can be seen on the graph of a quadratic where the graph crosses the -axis. Recall that the general quadratic equation: , can have two distinct roots, one repeated root or no roots at all (unless you are working with complex numbers).

Here are three examples that illustrate how quadratics can have different numbers of roots:

Note that we use a negative quadraticย to illustrate that a quadratic may have no roots but it is also possible for a positive quadratic to have no roots as well.

## The Value of the Discriminant

The discriminant can tell you how many roots a quadratic equation will have without having to actually find them. For the quadratic equation , the discriminant is given by .

Firstly, if , the equation has two distinct roots.

Alternatively, if , the equation has one repeated root.

Lastly, if , the equation has no roots.

See Examples and see discriminants for higher order polynomials

## Examples

## Videos

Finding the discriminant of a quadratic with algebraic coefficients and using the fact there are no roots to find the values for the unknown value.