# The Discriminant

## What are roots and the discriminant?

The **discriminant **of a quadratic equation will tell you how many roots the quadratic equation has.

Solutions of a quadratic equation are known as **roots**; they can be seen on the graph of a quadratic where the graph crosses the $x$-axis. Recall that the general quadratic equation: $ax^2+bx+c=0$, 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 quadratic 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 $ax^2+bx+c=0$, the discriminant is given by $b^2-4ac$.

**Firstly, if $b^2-4ac \hspace{2pt}>\hspace{2pt} 0$, the equation has two distinct roots.****Alternatively, if $b^2-4ac = 0$, the equation has one repeated root.****Lastly, if $b^2-4ac\hspace{2pt}<\hspace{2pt}0$, the equation has no roots.**

See Examples and see discriminants for higher order polynomials

## Examples

Given that the quadratic equation $kx^2-4x+2$ has equal roots, find the value of $k$.

If a quadratic equation has equal roots then $b^2-4ac=0$ where $a$, $b$ and $c$ are identified from the quadratic $ax^2+bx+c$. In this case, $a=k$, $b=-4$ and $c=2$. The discriminant is then $(-4)^2-4\times k\times 2=16-8k$. If this must be zero then the value of $k$ must be 2.

A quadratic has equation $y=px^2+3px-5$.

- Find an expression, in terms of $p$, for the discriminant of the quadratic.
- Given that the quadratic has two distinct roots, find the possible values of $p$.

- The discriminant is $b^2-4ac$ where $a=p$, $b=3p$ and $c=-5$. It follows that it is $(3p)^2-4\times p\times-5=9p^2+20p$.
- If the quadratic has two distinct roots, the discriminant must be positive, i.e. $ 9p^2+20p\hspace{2pt}>\hspace{2pt}0$ or $p(9p+20)\hspace{2pt}>\hspace{2pt}0$. In order for this inequality to hold we must have $p\hspace{2pt}>\hspace{2pt} 0 $ or $p\hspace{2pt}<\hspace{2pt}-\frac{20}{9}$ (you can see this by sketching $9p^2+20p$ and seeing where it is positive – see Inequalities).