Discriminants (DP IB Analysis & Approaches (AA)): Revision Note

Discriminants

What is the discriminant of a quadratic function?

• The discriminant of a quadratic is denoted by the Greek letter (upper case delta)

• For the quadratic function the discriminant is given by

  •  

    • This is given in the formula booklet

• The discriminant is the expression that is square rooted in the quadratic formula

How does the discriminant of a quadratic function affect its graph and roots?

• If then  and  are two distinct values

  • The equation  has two distinct real solutions

  • The graph of has two distinct real roots

    • This means the graph crosses the -axis twice

• If then and  are both zero

  • The equation has one repeated real solution

  • The graph of has one repeated real root

    • This means the graph touches the -axis at exactly one point

    • This means that the -axis is a tangent to the graph

• If then  and  are both undefined

  • The equation has no real solutions

  • The graph of has no real roots

    • This means the graph never touches the x-axis

    • This means that graph is wholly above (or wholly below) the -axis

Forming equations and inequalities using the discriminant

• A common question is to find an unknown coefficient in a quadratic

  • Questions usually use the letter for the unknown constant

• You will be given a fact about the quadratic such as:

  • The number of solutions of the equation

  • The number of roots of the graph

• To find the value or range of values of

  • Find an expression for the discriminant using 

  • Decide whether , or

    • If the question says there are real roots but does not specify how many then use

  • Solve the resulting equation or inequality to find the value(s) of