Discriminant (SQA National 5 Maths): Revision Note

Using the discriminant to determine the nature of roots

What is the discriminant?

• Recall the quadratic formula $x=\frac{-b\pm \sqrt{\left(b^2-4ac\right)}}{2a}$

  • which gives the solutions to the quadratic equation $a{x}^2+bx+c=0$

• The part of the formula under the square root, b² – 4ac, is called the discriminant

How can I use the discriminant to determine the nature of the roots of a quadratic function?

• A quadratic function is a function that can be written in the form $f\left(x\right)=a{x}^2+bx+c$, where $a\ne 0$

  • The roots of a quadratic function are the solutions to the equation $f\left(x\right)=0$

    • I.e. to the quadratic equation $a{x}^2+bx+c=0$

• The sign of the discriminant tells you about the roots of the quadratic function

  • If b² – 4ac > 0 (positive)

    • then the function has two real and distinct roots

      • i.e. the equation has two solutions, and they are different from each other

  • If b² – 4ac = 0 

    • then the function has one repeated real root (which may also be described as two equal real roots)

      • i.e. the equation only has one distinct solution (which may also be seen as it having two solutions which are the same)

  • If b² – 4ac < 0 (negative)

    • then the function has no real roots

      • i.e. the equation has no real number solutions

• Interestingly, if b² – 4ac is a perfect square number ( 1, 4, 9, 16, …) then this tells you that the quadratic expression ax² + bx + c can be factorised!