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

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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

    • straight capital delta equals b squared minus 4 a c 

      • 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 square root of b squared minus 4 a c end root and negative square root of b squared minus 4 a c end root are two distinct values

    • The equation a x squared plus b x plus c equals 0 has two distinct real solutions

    • The graph of space y equals a x squared plus b x plus c has two distinct real roots

      • This means the graph crosses the x-axis twice

  • If Δ = 0 then square root of b squared minus 4 a c end root and negative square root of b squared minus 4 a c end root are both zero

    • The equation a x squared plus b x plus c equals 0 has one repeated real solution

    • The graph of space y equals a x squared plus b x plus c 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 square root of b squared minus 4 a c end root and negative square root of b squared minus 4 a c end root are both undefined

    • The equation a x squared plus b x plus c equals 0 has no real solutions

    • The graph of space y equals a x squared plus b x plus c 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

Discrimamts Notes Diagram 2

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 straight capital delta equals b squared minus 4 a c

    • 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

Examiner Tips and Tricks

  • Questions will rarely use the word discriminant so it is important to recognise when its use is required

    • Look for

      • a number of roots or solutions being stated

      • whether and/or how often the graph of a quadratic function intercepts the x-axis

  • Be careful setting up inequalities that concern "two real roots" (increment greater or equal than 0) as opposed to "two real distinct roots" (increment greater than 0)

Worked Example

A function is given by space f left parenthesis x right parenthesis equals 2 k x squared plus k x minus k plus 2 , where k is a constant. The graph of space y equals f left parenthesis x right parenthesis has two distinct real roots.

a) Show that 9 k squared minus 16 k greater than 0.

2-2-5-ib-aa-sl-discriminant-a-we-solution

b) Hence find the set of possible values of k.

2-2-5-ib-aa-sl-discriminant-b-we-solution
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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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