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medium importancemedium difficulty7 min

Repeated roots

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

In one line: Zero discriminant, same answer twice.

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Understand it

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.

Example

x2 - 6x + 9 = 0 gives the root x = 3 twice.

Formula and rule

If D = 0, then the quadratic equation has equal roots.

The repeated root can often be written as -b / 2a.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

For x2 - 6x + 9 = 0, factorisation gives (x - 3)2 = 0.

  1. 1So x = 3 and x = 3.
  2. 2Since both roots are the same, they are repeated roots.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

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

Common wrong answer

Saying the roots are 3 and -3.

Why it loses marks: The factorisation gives only one repeated value.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

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.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    Example: x2 - 6x + 9 = 0 gives the root x = 3 twice.

  3. 3

    Add for 3 marks

    When D = 0, the quadratic just touches the x-axis at one point if drawn as a graph.

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