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Repeated roots

Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero.

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Student-friendly explanation

When D = 0, the quadratic just touches the x-axis at one point if drawn as a graph. Algebraically, this means the same root appears twice.

How to write this in exams

  1. 1

    Start with the exact idea

    Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero.

  2. 2

    Then show how to use it

    1. Find D or factorise the equation. 2. Check whether D = 0. 3. State that the roots are equal. 4. If required, solve to find the repeated value.

  3. 3

    Add one concrete example

    x^2 - 6x + 9 = 0 gives the root x = 3 twice.

  4. 4

    Avoid this incomplete answer

    Saying the roots are 3 and -3 is wrong because the factorisation gives only one repeated value.

Definition

Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero.

Example

x^2 - 6x + 9 = 0 gives the root x = 3 twice.

Rule to remember

If D = 0, then the quadratic equation has equal roots. The repeated root can often be written as -b / 2a.

Memory hook

Zero discriminant, same answer twice.

Examples and method

Worked example

For x^2 - 6x + 9 = 0, factorisation gives (x - 3)^2 = 0. So x = 3 and x = 3. Since both roots are the same, they are repeated roots.

Method to apply

1. Find D or factorise the equation. 2. Check whether D = 0. 3. State that the roots are equal. 4. If required, solve to find the repeated value.

Diagram support

A graph sketch touching the x-axis at one point helps show repeated roots clearly.

How CBSE asks it

Questions may ask for the nature of roots when D = 0 or ask for the repeated root of a simple equation.

Avoid common mistakes

Common confusion

Students may write two different values even when the discriminant is zero.

Common wrong answer

Saying the roots are 3 and -3 is wrong because the factorisation gives only one repeated value.

Exam tip

Whenever D = 0, state that the roots are real and equal, not just real.

Quick check

What does D = 0 tell you about the roots of a quadratic equation?

It tells us that the roots are equal and real. In other words, the same root occurs twice, so the equation has repeated roots.

Answer writing and exam use

1-mark answer

Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero.

2-mark answer

Repeated roots are two equal roots of a quadratic equation, and they occur when the discriminant is zero. If D = 0, then the quadratic equation has equal roots. The repeated root can often be written as -b / 2a. x^2 - 6x + 9 = 0 gives the root x = 3 twice.

3-mark answer

When D = 0, the quadratic just touches the x-axis at one point if drawn as a graph. Algebraically, this means the same root appears twice. If D = 0, then the quadratic equation has equal roots. The repeated root can often be written as -b / 2a. For x^2 - 6x + 9 = 0, factorisation gives (x - 3)^2 = 0. So x = 3 and x = 3. Since both roots are the same, they are repeated roots. Questions may ask for the nature of roots when D = 0 or ask for the repeated root of a simple equation. Saying the roots are 3 and -3 is wrong because the factorisation gives only one repeated value.
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