The discriminant of a quadratic equation

The discriminant of a quadratic equation is calculated using the part under the square root in the quadratic formula.

D=b^2-4ac

where a, b and c are the coefficients of the quadratic equation in the form ax^2+bx+c=0.

Symbol for the discriminant

The discriminant is usually represented by the greek letter *delta$ (\Delta). In this chapter, we will refer to it by the letter D.

Uses of the discriminant

The discriminant tells us the number of real roots that a quadratic equation has.
This is really useful, because it can be used to tell us whether it is possible to solve an equation or not.

It can also be used to find a constant in a quadratic equation (e.g. to find k in the equation x^2+kx+4=0), if we are given that, for example, the equation has one real root.

Finding the number of roots of an equation

We can find the number of real roots by working out the value of the discrimant. We can do that by calculating D=b^2-4ac.

• If D>0, there are two distinct real roots.
• If D=0, there is one real root (a repeated root).
• If D<0, there are no real roots (the roots are all complex/imaginary).

Example: find the number of real roots of the equation 2x^2-4x+1=0.

• a=2, b=-4 and c=1.
• Calculate the discriminant:
  • D=b^2-4ac
  • D=(-4)^2-4\times2\times1
  • D=16-8
  • D=8
• Because D>0, there are two distinct real roots.

Finding a constant using the discriminant

We can also use the discriminant to find a constant in a quadratic equation (where one of a, b or c has a non-integer coefficient, such as k). For example, if we are given that a quadratic equation has one real root, we can set D=0 and solve for the constant.

Repeated roots

When a quadratic equation has one real root (i.e. when D=0), this is called a repeated root.
This means that both roots of the equation are the same (i.e. they are equal) - and so we kind of only have one solution instead of the usual two.