Search
StudyWell
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account

The Discriminant

StudyWell > Algebra and Functions in A-Level Maths > The Discriminant

What are roots and the discriminant?

The discriminant of a quadratic equation will tell you how many roots the quadratic equation has.

Solutions of a quadratic equation are known as roots; they can be seen on the graph of a quadratic where the graph crosses the $x$-axis. Recall that the general quadratic equation: $ax^2+bx+c=0$, can have two distinct roots, one repeated root or no roots at all (unless you are working with complex numbers).

Here are three examples that illustrate how quadratics can have different numbers of roots:

discriminant

2 distinct roots

discriminant

1 repeated root

discriminant

no roots

Note that we use a negative quadratic  to illustrate that a quadratic may have no roots but it is also possible for a positive quadratic to have no roots as well.

The Value of the Discriminant

The discriminant can tell you how many roots a quadratic equation will have without having to actually find them.

For the quadratic equation $ax^2+bx+c=0$, the discriminant is given by $b^2-4ac$.

  • Firstly, if $b^2-4ac \hspace{2pt}>\hspace{2pt} 0$, the equation has two distinct roots.
  • Alternatively, if $b^2-4ac = 0$, the equation has one repeated root.
  • Lastly, if $b^2-4ac\hspace{2pt}<\hspace{2pt}0$, the equation has no roots.

See Examples and see discriminants for higher order polynomials

Examples

Example 1

Given that the quadratic equation $kx^2-4x+2$ has equal roots, find the value of $k$.

Show Solution

If a quadratic equation has equal roots then $b^2-4ac=0$ where $a$, $b$ and $c$ are identified from the quadratic $ax^2+bx+c$. In this case, $a=k$, $b=-4$ and $c=2$. The discriminant is then $(-4)^2-4\times k\times 2=16-8k$. If this must be zero then the value of $k$ must be 2.
Example 2

A quadratic has equation $y=px^2+3px-5$.

  1. Find an expression, in terms of $p$, for the discriminant of the quadratic.
  2. Given that the quadratic has two distinct roots, find the possible values of $p$.
Show Solution

  1. The discriminant is $b^2-4ac$ where $a=p$, $b=3p$ and $c=-5$. It follows that it is $(3p)^2-4\times p\times-5=9p^2+20p$.
  2. If the quadratic has two distinct roots, the discriminant must be positive, i.e. $ 9p^2+20p\hspace{2pt}>\hspace{2pt}0$ or $p(9p+20)\hspace{2pt}>\hspace{2pt}0$. In order for this inequality to hold we must have $p\hspace{2pt}>\hspace{2pt} 0 $ or $p\hspace{2pt}<\hspace{2pt}-\frac{20}{9}$ (you can see this by sketching $9p^2+20p$ and seeing where it is positive – see Inequalities).

Questions by Topic

DiscriminantsExamQuestions
  • Open Discriminant Questions by Topic in New Window
  • See more Questions by Topic

Videos

AS Maths Discriminants

See more videos

About

StudyWell is a website for students studying A-Level Maths (or equivalent. course). We have lots of resources including A-Level content delivered in manageable bite-size pieces, practice papers, past papers, questions by topic, worksheets, hints, tips, advice and much, much more.

Quick Links

  • CONTACT US
  • REGISTER
  • Edexcel Exam Timetable
  • Edexcel Formula Booklet
  • Edexcel Grade Boundaries
  • Edexcel Large Data Set
  • Edexcel Specification

Top Pages

  • A2 Maths (second year of A-Level Maths)
  • AS Maths (first year of A-Level Mathematics)
  • Blog
  • My account
  • Practice Papers
  • Questions by Topic
  • Shop
  • Membership Levels

Useful Websites

  • DESMOS
  • GeoGebra
  • Maths Challenges
  • STEP papers
  • UCAS
  • Wolfram Alpha
  • Friend of StudyWell: Elite Locksmiths
Footer logo
Copyright © 2022 StudyWell
MENU logo
  • Home
  • Maths
    • AS Maths
    • A2 Maths
    • Pure Maths
    • Statistics
    • Mechanics
  • Study Resources
    • Questions by Topic
    • Past & Practice Papers
    • AS Pure Maths Videos
  • Shop
  • My Account