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Completing the square

Completing the square means rewriting a quadratic expression so that part of it becomes a perfect square trinomial.

In one line: Half the x term, then square it.

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Understand it

This method is useful when factorisation is not easy. By adding and subtracting the right value, the quadratic can be written in a form that is easier to solve or analyse.

Example

x2 + 6x + 5 can be written as (x + 3)2 - 4.

Formula and rule

x2 + bx + (b/2)2 = (x + b/2)2.

If you add the square, you must also subtract it or keep the equation balanced.

See it worked out

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

Worked example

Solve x2 + 6x + 5 = 0 by completing the square.

  1. 1Write x2 + 6x = -5.
  2. 2Add 9 on both sides to get x2 + 6x + 9 = 4, so (x + 3)2 = 4.
  3. 3Then x + 3 = ±2, giving x = -1 or x = -5.

Method you can reuse

  1. 1Keep the x2 and x terms together.
  2. 2Take half of the x coefficient.
  3. 3Square that half.
  4. 4Add and subtract carefully or keep both sides balanced.
  5. 5Write the expression as a square plus or minus a constant.

Visual tip

A box model or square-grid sketch helps show why the added term creates a perfect square.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What number should be added to x2 + 10x to complete the square?

Think first, then show answer
Add 25, because half of 10 is 5 and 52 = 25. Then x2 + 10x + 25 becomes a perfect square trinomial.

Common wrong answer

Adding 5 instead of 25 to x2 + 10x.

Why it loses marks: The square of half the x coefficient is required.

Don't mix this up

Students often add half of b instead of adding the square of half of b.

Exam tip

For x2 + bx, add (b/2)2 to complete the square, then keep the equation balanced.

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

Students may be asked to complete the square, convert an expression into square form, or solve a quadratic using this method.
  1. 1

    Enough for 1 mark

    Completing the square means rewriting a quadratic expression so that part of it becomes a perfect square trinomial.

  2. 2

    Add for 2 marks

    Example: x2 + 6x + 5 can be written as (x + 3)2 - 4.

  3. 3

    Add for 3 marks

    This method is useful when factorisation is not easy.

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