C
CraftExam
high importancemedium8 min

Completing the square

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

Practice This Concept

Learn the concept

Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Adding 5 instead of 25 to x^2 + 10x is wrong because the square of half the x coefficient is required.

Definition

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

Example

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

Rule to remember

x^2 + bx + (b/2)^2 = (x + b/2)^2. If you add the square, you must also subtract it or keep the equation balanced.

Memory hook

Half the x term, then square it.

Examples and method

Worked example

Solve x^2 + 6x + 5 = 0 by completing the square. Write x^2 + 6x = -5. Add 9 on both sides to get x^2 + 6x + 9 = 4, so (x + 3)^2 = 4. Then x + 3 = ±2, giving x = -1 or x = -5.

Method to apply

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

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

Adding 5 instead of 25 to x^2 + 10x is wrong because the square of half the x coefficient is required.

Exam tip

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

Quick check

What number should be added to x^2 + 10x to complete the square?

Add 25, because half of 10 is 5 and 5^2 = 25. Then x^2 + 10x + 25 becomes a perfect square trinomial.

Answer writing and exam use

1-mark answer

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

2-mark answer

Completing the square means rewriting a quadratic expression so that part of it becomes a perfect square trinomial. x^2 + bx + (b/2)^2 = (x + b/2)^2. If you add the square, you must also subtract it or keep the equation balanced. x^2 + 6x + 5 can be written as (x + 3)^2 - 4.

3-mark answer

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. x^2 + bx + (b/2)^2 = (x + b/2)^2. If you add the square, you must also subtract it or keep the equation balanced. Solve x^2 + 6x + 5 = 0 by completing the square. Write x^2 + 6x = -5. Add 9 on both sides to get x^2 + 6x + 9 = 4, so (x + 3)^2 = 4. Then x + 3 = ±2, giving x = -1 or x = -5. Students may be asked to complete the square, convert an expression into square form, or solve a quadratic using this method. Adding 5 instead of 25 to x^2 + 10x is wrong because the square of half the x coefficient is required.
MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept