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16.1 Expanding Double Brackets

GCSE Maths Foundation · Edexcel · Chapter 16: Quadratic Equations and Graphs

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: multiplying out two brackets with the grid method and term by term, including squaring a bracket. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Expand two brackets such as (x + 3)(x − 5) and simplify
  • Square a bracket, such as (x + 4)², correctly
  • Expand brackets with a number in front of x, such as (2x + 1)(x + 3)
  • Use double brackets in area problems and "show that" questions

Key words

  • Expand: Multiply out the brackets.
  • Term: One part of an expression, e.g. x², 7x or −10.
  • Like terms: Terms with the same letters and powers, e.g. 3x and −5x.
  • Quadratic expression: An expression whose highest power is x², e.g. x² + 4x − 12.

The grid method

Every term in the first bracket multiplies every term in the second bracket. A grid makes sure you do not miss any. It is like finding the area of a rectangle split into four parts.

Worked example: (x + 5)(x + 3)

The four parts are x × x = x², x × 5 = 5x, 3 × x = 3x and 3 × 5 = 15.
Add them and collect the like terms: x² + 5x + 3x + 15.

Answer: x² + 8x + 15

Key fact: For (x + a)(x + b): the x term is (a + b)x and the number is a × b. In the example, 5 + 3 = 8 and 5 × 3 = 15.

Brackets with negative numbers

Keep the sign with each number. Multiply First, Outside, Inside, Last terms, or use a grid with the negative numbers written in.

  • x; x x²; −6 −6x
  • +4; x +4x; −6 −24

Worked example: (x + 4)(x − 6)

x² − 6x + 4x − 24
Collect like terms: −6x + 4x = −2x.

Answer: x² − 2x − 24

Worked example: (x − 3)(x − 7)

x² − 7x − 3x + 21 (negative × negative = positive)

Answer: x² − 10x + 21

Squaring a bracket

(x + 5)² means (x + 5)(x + 5). Write it out as two brackets first.

Worked example: squaring

(x + 5)² = (x + 5)(x + 5) = x² + 5x + 5x + 25 = x² + 10x + 25
(x − 4)² = (x − 4)(x − 4) = x² − 4x − 4x + 16 = x² − 8x + 16

Answer: x² + 10x + 25 and x² − 8x + 16

Key fact: (x + 5)² is not x² + 25. Check with x = 1: (1 + 5)² = 36, but 1 + 25 = 26.

Worked example: difference of two squares

(x + 6)(x − 6) = x² − 6x + 6x − 36. The x terms cancel.

Answer: x² − 36

Numbers in front of x and problems

Worked example: (2x + 1)(x + 4)

2x × x = 2x², 2x × 4 = 8x, 1 × x = x, 1 × 4 = 4

Answer: 2x² + 9x + 4

Worked example: show that

Show that (x + 4)(x + 2) − x(x + 3) = 3x + 8.
(x + 4)(x + 2) = x² + 6x + 8 and x(x + 3) = x² + 3x.
x² + 6x + 8 − x² − 3x = 3x + 8

Answer: shown: the x² terms cancel to leave 3x + 8

Check any expansion by substituting a small number. With x = 1: (5)(3) − 1(4) = 11 and 3(1) + 8 = 11. ✓

Common mistakes

  • Only multiplying the first terms and the last terms: (x + 3)(x + 2) is not x² + 6.
  • Writing (x + 5)² = x² + 25. There are four terms before collecting.
  • Sign errors: −3 × −7 = +21, not −21.
  • Forgetting to subtract every term of the second expansion in a "show that" question.

Revision questions

Try these, then check your answers below each one.

Question 1. Expand and simplify (x + 4)(x + 6). (Choose from: x² + 10x + 24, x² + 24, x² + 10x + 10, x² + 24x + 10.)

Answer: x² + 10x + 24

Working: x² + 6x + 4x + 24 = x² + 10x + 24.

Question 2. True or false? (x + 3)² = x² + 9

Answer: False

Working: (x + 3)² = x² + 6x + 9. The 6x is missing, so it is false. (Check with x = 1: 16, not 10.)

Question 3. Expand (x - 7)². (Choose from: x² - 14x + 49, x² + 49, x² - 49, x² - 7x + 49.)

Answer: x² - 14x + 49

Working: x² - 7x - 7x + 49 = x² - 14x + 49.

Related lessons

Continue in the full course

This lesson is section 16.1 (EGCMF1601) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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