Teach Me Maths

Expanding Single Brackets

Question 11 mark

Expand 3(x+2).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply both terms by the number outside.

Worked solution

Expand 3(x+2).

  1. 3\times x=3x and 3\times2=6.
  2. Therefore 3(x+2)=3x+6.

Question 21 mark

Expand 4(5+y).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The outside multiplier applies to both terms, whichever order they are written in.

Worked solution

Expand 4(5+y).

  1. 4\times5=20 and 4\times y=4y.
  2. The expansion is 20+4y, which can also be written 4y+20.

Question 31 mark

Expand 5(a-3).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Treat the second term inside as negative 3.

Worked solution

Expand 5(a-3).

  1. 5\times a=5a and 5\times(-3)=-15.
  2. The expansion is 5a-15.

Question 41 mark

Expand 2(3x+4).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the two coefficients in the first product.

Worked solution

Expand 2(3x+4).

  1. 2\times3x=6x and 2\times4=8.
  2. The expansion is 6x+8.

Question 52 marks

Expand x(x+6).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The same letter multiplied by itself gives a square.

Worked solution

Expand x(x+6).

  1. x\times x=x^2 and x\times6=6x.
  2. The expansion is x^2+6x. The powers are different, so the terms stay separate.

Question 62 marks

Expand 3a(2a+1).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Multiply the whole outside term by each inside term.

Worked solution

Expand 3a(2a+1).

  1. 3a\times2a=6a^2. Also, 3a\times1=3a.
  2. The expansion is 6a^2+3a.

Question 71 mark

Which option is the fully expanded form of -(x+7)?

Hint

A minus sign on its own means multiply every term by negative 1.

Worked solution

Which option is the fully expanded form of -(x+7)?

  1. -x+7
  2. x-7
  3. -x-7
  4. -(x+7)
  1. (-1)\times x=-x and (-1)\times7=-7.
  2. The expansion is -x-7. The option -(x+7) still contains the bracket, so it is not an expanded form.

Question 82 marks

Expand -3(4-t).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Keep the negative multiplier with each multiplication.

Worked solution

Expand -3(4-t).

  1. (-3)\times4=-12 and (-3)\times(-t)=3t.
  2. The expansion is -12+3t, or 3t-12.

Question 92 marks

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

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Expand the bracket, then collect the constants.

Worked solution

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

  1. Expanding gives 2x+6+5.
  2. Collect the constants: 6+5=11. The answer is 2x+11.

Question 102 marks

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

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

After expanding, remember the x outside the bracket.

Worked solution

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

  1. Expanding gives 4x+8+x.
  2. Collect 4x+x=5x. The answer is 5x+8.

Question 112 marks

Expand and simplify 3(2y-4)-y.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Expand first, then combine the y terms with their signs.

Worked solution

Expand and simplify 3(2y-4)-y.

  1. The expansion is 6y-12-y.
  2. 6y-y=5y, so the answer is 5y-12.

Question 122 marks

Expand and simplify 2(p+5)+3(2-p).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Expand both brackets before collecting. The letter is the second term in one bracket.

Worked solution

Expand and simplify 2(p+5)+3(2-p).

  1. The two expansions are 2p+10 and 6-3p.
  2. Together: 2p+10+6-3p.
  3. The letter terms give 2p-3p=-p and the constants give 10+6=16, so the answer is 16-p.

Question 132 marks

Expand and simplify 5(a+1)-2(a+4).

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Use negative 2 as the multiplier of the second bracket.

Worked solution

Expand and simplify 5(a+1)-2(a+4).

  1. The expression expands to 5a+5-2a-8.
  2. Collect: 5a-2a=3a and 5-8=-3. The answer is 3a-3.

Question 141 mark

Hana says 2(x+5)=2x+5. Write the correct expanded expression.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Check whether the outside multiplier has been applied to both terms.

Worked solution

Hana says 2(x+5)=2x+5. Write the correct expanded expression.

  1. The x term becomes 2x. The 5 must also be doubled, giving 10.
  2. The correct expansion is 2x+10. Hana multiplied only the first term.

Question 152 marks

The diagram shows a rectangle. All measurements are in centimetres.

Find a simplified expression for its perimeter.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

There are two lengths and two widths.

Worked solution

The diagram shows a rectangle. All measurements are in centimetres.

Find a simplified expression for its perimeter.

  1. The perimeter is 2(2x+1)+2(x+4).
  2. Expanding gives 4x+2+2x+8. Collecting gives (6x+10) cm.

Question 16Challenge2 marks

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

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Include the subtraction sign in the second multiplier.

Worked solution

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

  1. The first contribution is 6x-15. The second is -2x-8.
  2. Together: 6x-15-2x-8=4x-23.

Question 17Challenge3 marks

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

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

The outside term includes a negative sign and an x.

Worked solution

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

  1. The two products are (-2x)\times3x=-6x^2 and (-2x)\times(-4)=8x.
  2. Include the final 5x: -6x^2+8x+5x=-6x^2+13x.

Question 18Challenge3 marks

A rectangular corner is removed from a rectangular piece of card, as shown. All measurements are in centimetres.

Find an expanded, simplified expression for the area of the card remaining.

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Hint

Find each rectangular area using one bracket, then subtract the removed area.

Worked solution

A rectangular corner is removed from a rectangular piece of card, as shown. All measurements are in centimetres.

Find an expanded, simplified expression for the area of the card remaining.

  1. The original area is 4(3x+2)=12x+8.
  2. The removed area is 2(x+1)=2x+2.
  3. Subtract all of the removed area: 12x+8-2x-2=10x+6. The remaining area is (10x+6)\text{ cm}^2.

Question 19Challenge3 marks

Here is an expression:

3(2x-3)+4(x+1)

(a)

Expand and simplify the expression.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Work out the value of the expression when x=4.

1 mark

Hint

Multiply every term inside each bracket by the number outside it, then collect the x terms and the number terms.

Worked solution

Here is an expression:

3(2x-3)+4(x+1)

Part (a)

Expand and simplify the expression.

  1. 3(2x-3)=6x-9 and 4(x+1)=4x+4.
  2. Collect like terms: 6x+4x=10x and -9+4=-5, so the expression simplifies to 10x-5.

Part (b)

Work out the value of the expression when x=4.

  1. Use the simplified form: 10\times4-5=40-5=35.
  2. Check with the original: 3(8-3)+4(4+1)=15+20=35.

Question 20Challenge3 marks

Pack A contains (3n+4) stickers. Pack B contains (2n+4) stickers.

For any positive whole number n, how many more stickers are in 3 packs of B than in 2 packs of A?

Hint

Expand the two totals before comparing them; some terms will cancel.

Worked solution

Pack A contains (3n+4) stickers. Pack B contains (2n+4) stickers.

For any positive whole number n, how many more stickers are in 3 packs of B than in 2 packs of A?

  1. Two packs of A contain 2(3n+4)=6n+8 stickers.
  2. Three packs of B contain 3(2n+4)=6n+12 stickers.
  3. Both totals contain the same 6n. The difference is 12-8=4, so the three packs of B always contain 4 more stickers.

More on this topic: Expanding Single Brackets worksheet with full solutions

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