Discriminants (Cambridge International A-Level Mathematics)

Discriminants

What is a discriminant?

• The discriminant is the part of the quadratic formula that is under the square root sign $b^{2}-4ac$
• It is sometimes denoted by the Greek letter $Δ$ (capital delta)

How does the discriminant relate to graphs and roots?

There are three options for the outcome of the discriminant:

• If $b^{2}-4ac>0$ the quadratic crosses the x-axis twice meaning there are two distinct real roots
• If $b^{2}-4ac=0$ the quadratic touches the x-axis once meaning there is one real root (also called repeated roots)
• If $b^{2}-4ac<0$ the quadratic does not cross the x-axis meaning there are no real roots

Discriminant and inequalities

• You need to be able to set up and solve equations and inequalities (often quadratic) arising from the discriminant
• Sketch the quadratic and decide whether you're looking above or below zero to write your solutions correctly

Exam Hint

• When questions just mention “real roots”, the roots could be distinct or repeated (i.e. they aren't talking about complex numbers!)
• In these cases, you only need to worry about solving $b^{2}-4ac\geq0$
• When solving using inequalities, always sketch the quadratic and decide whether you're looking above or below zero to help write your solutions correctly

Worked Example

$k=0$ is excluded here because at $k=0$, the equation becomes $2=0$ which has no roots.