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 Δ (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 Δ > 0 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 x-axis twice

• If Δ = 0 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 x-axis at exactly one point

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

• If Δ < 0 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 below) the x-axis

Forming equations and inequalities using the discriminant

• Often at least one of the coefficients of a quadratic is unknown

  • Questions usually use the letter k 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 k

  • Find an expression for the discriminant

    • Use 

  • Decide whether Δ > 0, Δ = 0 or Δ < 0

    • If the question says there are real roots but does not specify how many then use Δ ≥ 0

  • Solve the resulting equation or inequality