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Expanding brackets

Single and double brackets. Topic 3 of 27 in Algebra.

Foundation & Higher This topic appears on both Foundation and Higher papers.

Video coming soon The written summary and downloads below are complete and ready to use.

Expanding brackets means multiplying them out. You multiply everything inside a bracket by what is outside it, and the brackets disappear. You will use it all through GCSE algebra: before you can simplify, solve or rearrange, there are often brackets to expand first. This lesson covers:

  • what expanding means, and why it works (the area model)
  • multiplying terms: numbers, letters, powers and signs
  • single brackets, with a number, a negative or a letter outside
  • expanding two or more brackets, then simplifying
  • double brackets such as (x + 4)(x − 2), using the grid method
  • squaring a bracket, such as (x + 5)2
  • the other ways an exam asks for it: areas, missing numbers, “show that” and number puzzles

Before you start, make sure you can collect like terms (that is lesson 2, Collecting like terms) and multiply negative numbers.

In the examples, the term outside a bracket is picked out like this while it multiplies. When terms are collected at the end, like terms share a colour and an underline, as in lesson 2: one group another group a third group numbers on their own

1. What does expanding mean?

Key idea To expand a bracket, multiply every term inside by the term outside. The answer has no brackets.

3(x + 2) means 3 lots of (x + 2). Write the three lots out and collect them:

3(x + 2) = (x + 2) + (x + 2) + (x + 2) = 3x + 6

You can also see it as an area. A rectangle 3 high and x + 2 long splits into two smaller rectangles. Each piece is one multiplication: 3 × x = 3x, and 3 × 2 = 6.

3x63x2
  • Expand, multiply out and remove the brackets all ask for the same thing.
  • Expand and simplify means expand, then collect any like terms.
  • Going the other way, from 3x + 6 back to 3(x + 2), is called factorising. That is lesson 4.

3(x + 2) ≡ 3x + 6 is an identity: it is true whatever number x is. Try x = 5: 3 × 7 = 21, and 15 + 6 = 21.

Watch out 3(x + 2) is not 3x + 2. The 3 multiplies the 2 as well. Multiplying only the first term is the most common slip in this whole topic.

Example 1: your first expansion

Expand 4(y + 3).

  1. Multiply 4 by each term inside the bracket
    4 × y = 4y
    4 × 3 = 12
Answer4y + 12

4 lots of (y + 3) is four y’s and four 3s: 4y + 12.

2. Multiplying terms

Key idea Every expansion is a set of small multiplications. Multiply the numbers, then write the letters next to each other. A letter times itself is squared: x × x = x2.
MultiplyThinkAnswer
5 × 3x5 × 3 = 1515x
x × 4The number goes in front.4x
x × yLetters side by side, in alphabetical order.xy
x × xx times itself.x2
2x × 3x2 × 3 = 6, and x × x = x26x2
3a × 4b3 × 4 = 12, and a × b = ab12ab
x × x2x × x × x: three x’sx3
2x × 5xy2 × 5 = 10, and x × x × y10x2y

Here powers such as x × x2 = x3 are worked out by counting the letters. The full rules for powers are in the Number topic Powers and roots.

The signs

Key idea Same signs give a positive. Different signs give a negative.
+ × + = +− × − = + + × − = −− × + = −

Example 2: signs, numbers and letters together

Work out (a) −4 × 2y    (b) 3p × −2p    (c) −a × −5b

QuestionSignNumbersLettersAnswer
(a) −4 × 2ydifferent: −4 × 2 = 8y−8y
(b) 3p × −2pdifferent: −3 × 2 = 6p × p = p2−6p2
(c) −a × −5bsame: +1 × 5 = 5ab5ab

−a means −1a, so its number is 1. Work out the sign first, then the numbers, then the letters.

Watch out x × x = x2, but x + x = 2x. Test it with x = 5: 5 × 5 = 25, but 2 × 5 = 10. And 3x × 3x = 9x2: the numbers multiply as well as the letters.

3. Expanding a single bracket

Key idea Multiply the term outside by each term inside, one at a time. Each term inside takes its sign into the multiplication with it.

Method

  1. Multiply the term outside by the first term inside
    Draw an arrow from the outside term to it, if that helps you keep track.
  2. Multiply the term outside by the next term inside, and the next
    Keep going until every term inside has been multiplied. Two terms inside means two multiplications; three means three.
  3. Write the answers down in order, each with its sign

Example 3: a number outside, and a minus inside

Expand 5(2y − 3).

  1. Multiply 5 by each term inside the bracket
    5 × 2y = 10y
    5 × (−3) = −15
Answer10y − 15

Example 4: a negative number outside

Expand −2(x − 6).

  1. Multiply −2 by each term inside the bracket
    −2 × x = −2x
    −2 × (−6) = +12
Answer−2x + 12

−2 × −6: the signs are the same, so the answer is positive. A negative outside changes the sign of every term inside.

Example 5: just a minus sign outside

Expand −(3x + 5).

  1. Multiply −1 by each term inside the bracket
    −1 × 3x = −3x
    −1 × 5 = −5
Answer−3x − 5

A minus sign on its own in front of a bracket means −1. Every sign inside changes.

Example 6: a letter outside

Expand x(x + 7).

  1. Multiply x by each term inside the bracket
    x × x = x2
    x × 7 = 7x
Answerx2 + 7x

x × x = x2, and x × 7 = 7x (the number goes in front). If a question writes the bracket first, (x + 7)x, it means the same.

Example 7: a number and a letter outside

Expand 4a(2a − 3b).

  1. Multiply 4a by each term inside the bracket
    4a × 2a = 8a2
    4a × (−3b) = −12ab
Answer8a2 − 12ab

Example 8: three terms inside

Expand 3(2x − y + 4).

  1. Multiply 3 by each term inside the bracket
    3 × 2x = 6x
    3 × (−y) = −3y
    3 × 4 = 12
Answer6x − 3y + 12

Three terms inside means three multiplications. −y is −1y, so 3 × −y = −3y.

Example 9: powers inside

Expand 2p(p2 + 3p − 5).

  1. Multiply 2p by each term inside the bracket
    2p × p2 = 2p3
    2p × 3p = 6p2
    2p × (−5) = −10p
Answer2p3 + 6p2 − 10p

p × p2 is p × p × p, which is p3.

Example 10: the number comes first inside

Expand 5(3 − 2x).

  1. Multiply 5 by each term inside the bracket
    5 × 3 = 15
    5 × (−2x) = −10x
Answer15 − 10x

You can also write it as −10x + 15. Both are right: each term keeps its own sign.

Example 11: a fraction outside

Expand 12(8x − 6).

  1. Multiply 12 by each term inside the bracket
    12 × 8x = 4x
    12 × (−6) = −3
Answer4x − 3

Multiplying by 12 is the same as halving: half of 8x is 4x, and half of 6 is 3.

Watch out A negative outside changes every sign inside. −2(x − 6) = −2x + 12. Writing −2x − 12 loses the mark.

4. Expand and simplify

Key idea Expand each bracket on its own, then collect like terms. The sign in front of a bracket belongs to the number outside it: in 5(x + 2) − 3(x + 4), the second bracket is multiplied by −3.

Method

  1. Expand each bracket
    Take the sign in front of each bracket with the number outside it.
  2. Write out every term
  3. Collect like terms
    Exactly as in lesson 2: add or subtract the numbers in front, and keep the letters.

Example 12: adding two brackets

Expand and simplify 3(x + 4) + 2(x + 1).

  1. Expand each bracket, with the sign in front of it
    3(x + 4) = 3x + 12
    2(x + 1) = 2x + 2
  2. Write out every term
    3x + 12 + 2x + 2
  3. Collect like terms
    3x + 2x = 5x
    12 + 2 = 14
Answer5x + 14

Example 13: a minus inside one bracket

Expand and simplify 4(2a + 3) + 5(a − 2).

  1. Expand each bracket, with the sign in front of it
    4(2a + 3) = 8a + 12
    5(a − 2) = 5a − 10
  2. Write out every term
    8a + 12 + 5a − 10
  3. Collect like terms
    8a + 5a = 13a
    12 − 10 = 2
Answer13a + 2

Example 14: subtracting a bracket

Expand and simplify 5(x + 2) − 3(x + 4).

  1. Expand each bracket, with the sign in front of it
    5(x + 2) = 5x + 10
    −3(x + 4) = −3x − 12
  2. Write out every term
    5x + 10 − 3x − 12
  3. Collect like terms
    5x − 3x = 2x
    10 − 12 = −2
Answer2x − 2

The second bracket is multiplied by −3, so −3 × 4 = −12.

Example 15: subtracting a bracket with a minus inside

Expand and simplify 6(y − 1) − 2(y − 5).

  1. Expand each bracket, with the sign in front of it
    6(y − 1) = 6y − 6
    −2(y − 5) = −2y + 10
  2. Write out every term
    6y − 6 − 2y + 10
  3. Collect like terms
    6y − 2y = 4y
    −6 + 10 = 4
Answer4y + 4

−2 × −5 = +10: two negatives make a positive.

Example 16: a minus sign in front of a bracket

Simplify 7x − (2x − 3).

  1. Expand each bracket, with the sign in front of it
    −(2x − 3) = −2x + 3
  2. Write out every term
    7x − 2x + 3
  3. Collect like terms
    7x − 2x = 5x
    3 has nothing to collect with
Answer5x + 3

The minus sign means −1 × the bracket, so both signs inside change.

Example 17: a letter outside one bracket

Expand and simplify x(x + 3) + 2(x − 5).

  1. Expand each bracket, with the sign in front of it
    x(x + 3) = x2 + 3x
    2(x − 5) = 2x − 10
  2. Write out every term
    x2 + 3x + 2x − 10
  3. Collect like terms
    x2 has nothing to collect with
    3x + 2x = 5x
    −10 has nothing to collect with
Answerx2 + 5x − 10

Example 18: letters outside both brackets

Expand and simplify x(2x − 1) − x(x − 4).

  1. Expand each bracket, with the sign in front of it
    x(2x − 1) = 2x2 − x
    −x(x − 4) = −x2 + 4x
  2. Write out every term
    2x2 − x − x2 + 4x
  3. Collect like terms
    2x2 − x2 = x2
    −x + 4x = 3x
Answerx2 + 3x

Example 19: two different letters

Expand and simplify 3(2a + b) − 2(a − 4b).

  1. Expand each bracket, with the sign in front of it
    3(2a + b) = 6a + 3b
    −2(a − 4b) = −2a + 8b
  2. Write out every term
    6a + 3b − 2a + 8b
  3. Collect like terms
    6a − 2a = 4a
    3b + 8b = 11b
Answer4a + 11b
Watch out Subtracting a bracket changes every sign inside it. In 5(x + 2) − 3(x + 4), the −3 multiplies the 4 as well: −3x − 12, not −3x + 12.

5. Double brackets

Key idea Multiply every term in the first bracket by every term in the second. Two terms times two terms gives four multiplications. Then collect like terms.

Why: (x + 2)(x + 3) is the area of a rectangle x + 2 high and x + 3 long. Build it from algebra tiles and count the pieces.

There is one x2 tile, 3 + 2 = 5 x tiles, and 2 × 3 = 6 one tiles. So (x + 2)(x + 3) = x2 + 5x + 6.

Method: the grid

  1. Draw a 2 by 2 grid
    Put the terms of one bracket down the side and the other along the top, each with its sign.
  2. Multiply to fill each box
    Each box is the term at the start of its row times the term at the top of its column.
  3. Write out the four terms and collect like terms
    Usually the two middle terms are like terms.

Example 20: two plus signs

Expand and simplify (x + 4)(x + 2).

  1. Multiply every term in one bracket by every term in the other
    ×x+2
    xx2+2x
    +4+4x+8
  2. Write out the four terms
    x2 + 2x + 4x + 8
  3. Collect like terms
    2x + 4x = 6x
Answerx2 + 6x + 8

Another way: FOIL. Multiply the First terms, the Outside terms, the Inside terms and the Last terms. It gives the same four products.

  • First: x × x = x2
  • Outside: x × 2 = 2x
  • Inside: 4 × x = 4x
  • Last: 4 × 2 = 8

Use whichever you find easier. The grid makes it harder to miss one out.

Example 21: one minus sign

Expand and simplify (x + 6)(x − 2).

  1. Multiply every term in one bracket by every term in the other
    ×x−2
    xx2−2x
    +6+6x−12
  2. Write out the four terms
    x2 − 2x + 6x − 12
  3. Collect like terms
    −2x + 6x = 4x
Answerx2 + 4x − 12

Example 22: two minus signs

Expand and simplify (x − 5)(x − 3).

  1. Multiply every term in one bracket by every term in the other
    ×x−3
    xx2−3x
    −5−5x+15
  2. Write out the four terms
    x2 − 3x − 5x + 15
  3. Collect like terms
    −3x − 5x = −8x
Answerx2 − 8x + 15

−5 × −3 = +15. Two negatives multiply to a positive.

Example 23: a number in front of x

Expand and simplify (2x + 3)(x − 4).

  1. Multiply every term in one bracket by every term in the other
    ×x−4
    2x2x2−8x
    +3+3x−12
  2. Write out the four terms
    2x2 − 8x + 3x − 12
  3. Collect like terms
    −8x + 3x = −5x
Answer2x2 − 5x − 12

Example 24: numbers in front of both x’s

Expand and simplify (3x − 2)(2x + 5).

  1. Multiply every term in one bracket by every term in the other
    ×2x+5
    3x6x2+15x
    −2−4x−10
  2. Write out the four terms
    6x2 + 15x − 4x − 10
  3. Collect like terms
    15x − 4x = 11x
Answer6x2 + 11x − 10

3x × 2x = 6x2: multiply the numbers, then x × x.

Example 25: when the middle terms cancel

Expand and simplify (x + 7)(x − 7).

  1. Multiply every term in one bracket by every term in the other
    ×x−7
    xx2−7x
    +7+7x−49
  2. Write out the four terms
    x2 − 7x + 7x − 49
  3. Collect like terms
    −7x + 7x = 0 they cancel out
Answerx2 − 49

The brackets are the same except for one sign, so the middle terms cancel and you are left with a square take away a square. This is called the difference of two squares.

Example 26: two different letters

Expand and simplify (a + 2b)(3a − b).

  1. Multiply every term in one bracket by every term in the other
    ×3a−b
    a3a2−ab
    +2b+6ab−2b2
  2. Write out the four terms
    3a2 − ab + 6ab − 2b2
  3. Collect like terms
    −ab + 6ab = 5ab
Answer3a2 + 5ab − 2b2

2b × 3a = 6ab. Write the letters in alphabetical order, so that −ab and 6ab are easy to spot as like terms.

Example 27: the number comes first

Expand and simplify (4 − x)(3 + x).

  1. Multiply every term in one bracket by every term in the other
    ×3+x
    412+4x
    −x−3x−x2
  2. Write out the four terms
    12 + 4x − 3x − x2
  3. Collect like terms
    4x − 3x = x
Answer−x2 + x + 12

−x × x = −x2. You can also write the answer as 12 + x − x2.

Watch out Four multiplications, every time. (x + 4)(x + 2) is not x2 + 8: that misses the two middle terms, 2x and 4x.

6. Squaring a bracket

Key idea (x + 3)2 means (x + 3)(x + 3). Write the bracket out twice, then expand it as a double bracket.

As an area, (x + 3)2 is a square with side x + 3. It has four pieces, not two:

x2 3xx3x93x3

So (x + 3)2 = x2 + 3x + 3x + 9 = x2 + 6x + 9. The two shaded 3x pieces are the part people forget when they write x2 + 9.

Example 28: a plus sign

Expand and simplify (x + 5)2.

  1. Write the bracket out twice
    (x + 5)2 = (x + 5)(x + 5)
  2. Multiply every term in one bracket by every term in the other
    ×x+5
    xx2+5x
    +5+5x+25
  3. Write out the four terms
    x2 + 5x + 5x + 25
  4. Collect like terms
    5x + 5x = 10x
Answerx2 + 10x + 25

Example 29: a minus sign

Expand and simplify (y − 4)2.

  1. Write the bracket out twice
    (y − 4)2 = (y − 4)(y − 4)
  2. Multiply every term in one bracket by every term in the other
    ×y−4
    yy2−4y
    −4−4y+16
  3. Write out the four terms
    y2 − 4y − 4y + 16
  4. Collect like terms
    −4y − 4y = −8y
Answery2 − 8y + 16

The last term is −4 × −4 = +16. A number squared is never negative.

Example 30: a number in front of x

Expand and simplify (3x − 2)2.

  1. Write the bracket out twice
    (3x − 2)2 = (3x − 2)(3x − 2)
  2. Multiply every term in one bracket by every term in the other
    ×3x−2
    3x9x2−6x
    −2−6x+4
  3. Write out the four terms
    9x2 − 6x − 6x + 4
  4. Collect like terms
    −6x − 6x = −12x
Answer9x2 − 12x + 4

3x × 3x = 9x2. The 3 is squared too, so it is not 3x2.

Example 31: a number outside a squared bracket

Expand and simplify 2(x + 1)2.

  1. Square the bracket first: powers come before multiplying
    (x + 1)2 = (x + 1)(x + 1) = x2 + x + x + 1 = x2 + 2x + 1
  2. Then multiply every term by 2
    2 × x2 = 2x2
    2 × 2x = 4x
    2 × 1 = 2
Answer2x2 + 4x + 2

Do not multiply the 2 into the bracket first: (2x + 2)2 would give 4x2 + 8x + 4, which is twice too big.

A pattern worth spotting. (x + a)2 = x2 + 2ax + a2. The middle term is double the number, times x. The last term is the number squared. For (x + 5)2: 2 × 5 = 10 and 52 = 25, giving x2 + 10x + 25. With a minus, (x − a)2 = x2 − 2ax + a2. If you are not sure, write the bracket out twice.

Watch out (x + 5)2 is not x2 + 25. Test it with x = 1: (1 + 5)2 = 36, but 1 + 25 = 26. The 10 that is missing is the two 5x pieces.

7. The other ways an exam asks for it

Key idea A question might not say “expand”. An area, a missing number, “show that” and a number puzzle can all need you to multiply out brackets.

Example 32: the area of a rectangle

A rectangle is 2x + 1 cm long and x + 4 cm wide. Write an expression for its area. Give your answer in its simplest form.

2x + 1 x + 4
  1. Area = length × width
    (2x + 1)(x + 4)
    Use brackets: the whole of each side multiplies.
  2. Multiply every term in one bracket by every term in the other
    ×x+4
    2x2x2+8x
    +1+x+4
  3. Write out the four terms
    2x2 + 8x + x + 4
  4. Collect like terms
    8x + x = 9x
Answer2x2 + 9x + 4 cm2

Example 33: a shaded area

A small rectangle is cut out of a large one. Write an expression for the shaded area, in its simplest form. The lengths are in centimetres.

x + 5 x + 3 x 2
  1. Shaded area = large rectangle − small rectangle
    (x + 5)(x + 3) − 2x
    The small rectangle is x by 2, so its area is 2x.
  2. Expand each bracket, with the sign in front of it
    (x + 5)(x + 3) = x2 + 3x + 5x + 15 = x2 + 8x + 15
  3. Write out every term
    x2 + 8x + 15 − 2x
  4. Collect like terms
    x2 has nothing to collect with
    8x − 2x = 6x
    15 has nothing to collect with
Answerx2 + 6x + 15 cm2

Example 34: show that

Show that (x + 4)(x + 1) − x(x + 3) ≡ 2x + 4.

  1. Expand each bracket, with the sign in front of it
    (x + 4)(x + 1) = x2 + x + 4x + 4 = x2 + 5x + 4
    −x(x + 3) = −x2 − 3x
  2. Write out every term
    x2 + 5x + 4 − x2 − 3x
  3. Collect like terms
    x2 − x2 = 0 they cancel out
    5x − 3x = 2x
    4 has nothing to collect with
Answer2x + 4

(x + 4)(x + 1) − x(x + 3) = 2x + 4, which is the right-hand side.

In a “show that” the answer is given, so the marks are for the working. Write every step, and finish by saying it matches.

Example 35: find the missing number

Fill in the box to make each one true.

QuestionThinkAnswer
4(x + □) = 4x + 284 × what = 28? 77
□(2x − 3) = 10x − 15what × 2x = 10x? 5. Check: 5 × −3 = −155
x(□ − 3) = 5x2 − 3xx × what = 5x2? 5x5x
(x + 3)(x + □) = x2 + 8x + 153 × what = 15? 5. Check: 3x + 5x = 8x5

Use one term to find the number, then check it with another term.

Example 36: a number puzzle

Think of a number. Add 3. Multiply by 4. Take away 12. Show that you always end with 4 times the number you thought of.

  1. Call the number n and follow the steps
    n → n + 3 → 4(n + 3) → 4(n + 3) − 12
  2. Expand the bracket
    4(n + 3) = 4n + 12
  3. Collect like terms
    4n + 12 − 12
    4n has nothing to collect with
    12 − 12 = 0 they cancel out
Answer4(n + 3) − 12 = 4n, which is 4 times the number

Example 37: check your answer with a number

Zoe expands (x + 3)(x − 2) and gets x2 + x − 6. Ravi gets x2 − 6. Check both with x = 4.

  • The question: (4 + 3)(4 − 2) = 7 × 2 = 14
  • Zoe: 42 + 4 − 6 = 16 + 4 − 6 = 14. It matches.
  • Ravi: 42 − 6 = 16 − 6 = 10. It does not match, so Ravi is wrong. He missed the middle terms.

A matching check is good evidence, not a proof. Use a number like 3 or 4, not 0 or 1, because 0 and 1 can hide mistakes.

The mistakes that cost marks

  1. Multiplying only the first term. 4(x + 5) written as 4x + 5. the term outside multiplies every term inside: 4x + 20.
  2. Not changing the signs when the outside is negative. −2(x − 6) written as −2x − 12. −2 × −6 = +12, so −2x + 12.
  3. Subtracting a bracket but changing only the first sign. 5(x + 2) − 3(x + 4) written as 5x + 10 − 3x + 12. the −3 multiplies both terms: −3x − 12.
  4. Writing x × x as 2x. x × x = x2. 2x is x + x.
  5. Forgetting to multiply the numbers as well. 3x × 3x written as 3x2. 3 × 3 = 9, so 9x2.
  6. Missing the middle terms. (x + 4)(x + 2) written as x2 + 8. four multiplications: x2 + 2x + 4x + 8 = x2 + 6x + 8.
  7. Squaring each term. (x + 5)2 written as x2 + 25. write it out twice, (x + 5)(x + 5), to get x2 + 10x + 25.
  8. A negative times a negative left negative. (x − 5)(x − 3) ending in −15. −5 × −3 = +15.
  9. Collecting unlike terms after expanding. x2 + 7x written as 8x2. x2 and x are different kinds of term, so x2 + 7x stays as it is.
  10. Multiplying into a squared bracket first. 2(x + 1)2 worked out as (2x + 2)2. square the bracket first, then multiply by 2: 2x2 + 4x + 2.

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