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GCSEMathsAQARevision Notes2. AlgebraExpanding BracketsExpanding Single Brackets

Expanding Single Brackets (AQA GCSE Maths)

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Mark

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Mark

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Expanding One Bracket

How do I expand a bracket?

  • The expression 3x(x + 2) means 3x  multiplied by the bracket (x + 2)
    • 3x is the term outside the bracket (sometimes called a "factor") and x + 2 are the terms inside the bracket
  • Expanding the brackets means multiplying the term on the outside by each term on the inside
    • This will remove / "get rid of" the brackets
    • 3x(x + 2) expands to 3 x cross times x plus 3 x cross times 2 which simplifies to 3 x squared plus 6 x

Beware of minus signs

  • Remember the basic rules of multiplication with signs
    •  −  ×  −  =  +
    • −  ×  +  =  − 
  • It helps to put brackets around negative terms

Worked example

(a)
Expand  4 x open parentheses 2 x minus 3 close parentheses.


Multiply the 4 x term outside the brackets by both terms inside the brackets, watch out for negatives!

4 x cross times 2 x plus 4 x cross times open parentheses negative 3 close parentheses

Simplify.

bold 8 bold italic x to the power of bold 2 bold minus bold 12 bold italic x

 

(b)
Expand  negative 7 x open parentheses 4 minus 5 x close parentheses.


Multiply the negative 7 x outside the brackets by both terms inside the brackets, watch out for negatives!

open parentheses negative 7 x close parentheses cross times 4 plus open parentheses negative 7 x close parentheses cross times open parentheses negative 5 x close parentheses

Simplify.

bold minus bold 28 bold italic x bold plus bold 35 bold italic x to the power of bold 2

Expand & Simplify

How do I simplify an expression when there is more than one term in brackets?

  • Look out for two or more terms that contain brackets in an expression that are being added/subtracted
    • E.g. 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses
    • Notice that the two sets of brackets are connected by a + sign, so you are not multiplying the brackets together
  • STEP 1: Expand each set of brackets separately by multiplying the term on the outside of the brackets by each of the terms on the inside, be careful with negative terms
    • E.g. the first set of brackets expands to 4 cross times x plus 4 cross times 7, and simplifies to 4 x plus 28, the second set of brackets expands to 5 x cross times 3 plus 5 x cross times open parentheses negative x close parentheses and simplifies to 15 x minus 5 x squared
    • So, 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses equals 4 x plus 28 plus 15 x minus 5 x squared
  • STEP 2: Collect together like terms
    • E.g. 4 open parentheses x plus 7 close parentheses plus 5 x open parentheses 3 minus x close parentheses equals 19 x plus 28 minus 5 x squared 

Worked example

(a)
Expand and simplify  2 open parentheses x plus 5 close parentheses plus 3 x open parentheses x minus 8 close parentheses.


Expand each set of brackets separately by multiplying the term outside the brackets by each of the terms inside the brackets.
Keep negative terms inside brackets so that you don't miss them!

2 cross times x plus 2 cross times 5 plus 3 x cross times x plus 3 x cross times open parentheses negative 8 close parentheses

Simplify.

2 x plus 10 plus 3 x squared minus 24 x

Collect 'like' terms.

bold minus bold 22 bold italic x bold plus bold 10 bold plus bold 3 bold italic x to the power of bold 2

(b)
Expand and simplify  3 x open parentheses x plus 2 close parentheses minus 7 open parentheses x minus 6 close parentheses.


Expand each set of brackets separately by multiplying the term outside the brackets by each of the terms inside the brackets.
Keep negative terms inside brackets so that you don't miss them!

3 x cross times x plus 3 x cross times 2 plus open parentheses negative 7 close parentheses cross times x plus open parentheses negative 7 close parentheses cross times open parentheses negative 6 close parentheses

Simplify.

3 x squared plus 6 x minus 7 x plus 42

Collect 'like' terms.

bold 3 bold italic x to the power of bold 2 bold minus bold italic x bold plus bold 42

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.