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
- $Δ = b^{2} - 4 a c$
  - 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 $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are two distinct values
- The equation $a x^{2} + b x + c = 0$ has two distinct real solutions
- The graph of $y = a x^{2} + b x + c$ has two distinct real roots
  - This means the graph crosses the x-axis twice
If Δ = 0 then $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are both zero
- The equation $a x^{2} + b x + c = 0$ has one repeated real solution
- The graph of $y = a x^{2} + b x + c$ 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 $\sqrt{b^{2} - 4 a c}$ and $- \sqrt{b^{2} - 4 a c}$ are both undefined
- The equation $a x^{2} + b x + c = 0$ has no real solutions
- The graph of $y = a x^{2} + b x + c$ 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

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 $Δ = b^{2} - 4 a c$
- 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 Tip

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" ( $∆ \geq 0$ ) as opposed to "two real distinct roots" ( $∆ > 0$ )

Worked example

A function is given by $f (x) = 2 k x^{2} + k x - k + 2$ , where $k$ is a constant. The graph of $y = f (x)$ has two distinct real roots.

Show that

9 k^{2} - 16 k > 0

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

Hence find the set of possible values of

k

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

Discriminants (DP IB Maths: AA HL)

Revision Note

What is the discriminant of a quadratic function?

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

Forming equations and inequalities using the discriminant

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities