Teach Me Maths

Expanding Double Brackets

Question 12 marks

Expand and simplify (x+3)(x+4)

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

Hint

Multiply each term in the first bracket by each term in the second bracket, then collect like terms.

Worked solution

Expand and simplify (x+3)(x+4)

  1. x\times x=x^2, x\times4=4x, 3\times x=3x and 3\times4=12.
  2. So (x+3)(x+4)=x^2+4x+3x+12.
  3. Collect the x terms: x^2+7x+12.

Question 21 mark

Which of these is (x+5)(x+2) expanded and simplified?

Hint

There should be four multiplications before you collect like terms.

Worked solution

Which of these is (x+5)(x+2) expanded and simplified?

  1. x^2+10
  2. 2x+7x+10
  3. x^2+7x+10
  4. x^2+7x+7
  1. x\times x=x^2, x\times2=2x, 5\times x=5x and 5\times2=10.
  2. x^2+2x+5x+10=x^2+7x+10.
  3. x^2+10 misses the two x terms, and x\times x is x^2, not 2x.

Question 32 marks

Expand and simplify (x+6)(x-2)

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

Hint

Keep the sign in front of each number with it when you multiply.

Worked solution

Expand and simplify (x+6)(x-2)

  1. x\times x=x^2, x\times(-2)=-2x, 6\times x=6x and 6\times(-2)=-12.
  2. So (x+6)(x-2)=x^2-2x+6x-12.
  3. -2x+6x=4x, so the answer is x^2+4x-12.

Question 42 marks

Expand and simplify (n-3)(n-5)

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

Hint

A negative number multiplied by a negative number gives a positive number.

Worked solution

Expand and simplify (n-3)(n-5)

  1. n\times n=n^2, n\times(-5)=-5n, -3\times n=-3n and -3\times(-5)=15.
  2. So (n-3)(n-5)=n^2-5n-3n+15.
  3. -5n-3n=-8n, so the answer is n^2-8n+15.

Question 52 marks

Here is a rectangle.

Find an expression, in terms of x, for the area of the rectangle. Give your answer without brackets, in its simplest form.

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

Hint

Area of a rectangle is length times width, so multiply the two brackets.

Worked solution

Here is a rectangle.

Find an expression, in terms of x, for the area of the rectangle. Give your answer without brackets, in its simplest form.

  1. Area =(x+4)(x+1).
  2. x\times x=x^2, x\times1=x, 4\times x=4x and 4\times1=4.
  3. x^2+x+4x+4=x^2+5x+4 cm².

Question 62 marks

Expand and simplify (x+5)^2

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

Hint

Write the square as (x+5)(x+5), then multiply every term in one bracket by every term in the other.

Worked solution

Expand and simplify (x+5)^2

  1. (x+5)^2=(x+5)(x+5).
  2. x\times x=x^2, x\times5=5x, 5\times x=5x and 5\times5=25.
  3. x^2+5x+5x+25=x^2+10x+25.
  4. It is not x^2+25: that misses the two 5x terms.

Question 72 marks

Expand and simplify (3+t)(t-8)

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

Hint

It does not matter that the number comes first in the first bracket: still do four multiplications.

Worked solution

Expand and simplify (3+t)(t-8)

  1. 3\times t=3t, 3\times(-8)=-24, t\times t=t^2 and t\times(-8)=-8t.
  2. So (3+t)(t-8)=3t-24+t^2-8t.
  3. 3t-8t=-5t, so the answer is t^2-5t-24.

Question 82 marks

Expand and simplify (x+7)(x-7)

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

Hint

Do all four multiplications and look carefully at the two x terms.

Worked solution

Expand and simplify (x+7)(x-7)

  1. x\times x=x^2, x\times(-7)=-7x, 7\times x=7x and 7\times(-7)=-49.
  2. x^2-7x+7x-49.
  3. -7x+7x=0, so the x terms cancel and the answer is x^2-49.

Question 92 marks

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

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

Hint

Remember that 2x\times x is 2x^2.

Worked solution

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

  1. 2x\times x=2x^2, 2x\times4=8x, 3\times x=3x and 3\times4=12.
  2. 2x^2+8x+3x+12.
  3. 8x+3x=11x, so the answer is 2x^2+11x+12.

Question 101 mark

Which of these is (x-4)^2 expanded and simplified?

Hint

Write it as (x-4)(x-4) and remember that a negative times a negative is positive.

Worked solution

Which of these is (x-4)^2 expanded and simplified?

  1. x^2-16
  2. x^2+16
  3. x^2-8x+16
  4. x^2-8x-16
  1. (x-4)^2=(x-4)(x-4).
  2. x\times x=x^2, x\times(-4)=-4x, -4\times x=-4x and -4\times(-4)=16.
  3. x^2-4x-4x+16=x^2-8x+16.
  4. x^2-16 and x^2+16 miss the x terms; -16 has the wrong sign.

Question 112 marks

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

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

Hint

Multiply the numbers in front of the xs too: 4x\times2x is 8x^2.

Worked solution

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

  1. 4x\times2x=8x^2, 4x\times(-1)=-4x, 3\times2x=6x and 3\times(-1)=-3.
  2. 8x^2-4x+6x-3.
  3. -4x+6x=2x, so the answer is 8x^2+2x-3.

Question 123 marks

Ben expands and simplifies (x-5)(x+2)

Here is his working.

(x-5)(x+2)=x^2+2x-5x+10

=x^2-3x+10

(a)

What mistake has Ben made?

1 mark

(b)

Expand and simplify (x-5)(x+2) correctly.

2 marks

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

Hint

Check each of Ben's four multiplications, including the signs.

Worked solution

Ben expands and simplifies (x-5)(x+2)

Here is his working.

(x-5)(x+2)=x^2+2x-5x+10

=x^2-3x+10

Part (a)

What mistake has Ben made?

  1. -5\times2 should be -10, not 10
  2. x\times x should be 2x, not x^2
  3. 2x-5x should be 3x, not -3x
  4. He should have added -5 and 2 to get the last number
  1. The four multiplications should be x^2, 2x, -5x and -5\times2=-10.
  2. Ben wrote +10, but a negative times a positive is negative.

Part (b)

Expand and simplify (x-5)(x+2) correctly.

  1. (x-5)(x+2)=x^2+2x-5x-10.
  2. 2x-5x=-3x, so the answer is x^2-3x-10.

Question 132 marks

Expand and simplify (x+4)(y-3)

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

Hint

Do all four multiplications as usual, then check whether any of the terms are like terms.

Worked solution

Expand and simplify (x+4)(y-3)

  1. x\times y=xy, x\times(-3)=-3x, 4\times y=4y and 4\times(-3)=-12.
  2. xy, -3x, 4y and -12 are all different kinds of term, so none of them can be collected.
  3. The answer is xy-3x+4y-12.

Question 142 marks

(x+4)(x+k)=x^2+11x+28

k is a number.

Work out the value of k.

Hint

Look at the number on the end: what do 4 and k multiply to give?

Worked solution

(x+4)(x+k)=x^2+11x+28

k is a number.

Work out the value of k.

  1. Expanding, the number on the end is 4\times k, so 4k=28 and k=7.
  2. Check the x term: 4x+7x=11x, which matches.
  3. k=7

Question 154 marks

Here is a rectangle.

All the lengths are in centimetres.

(a)

Find an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.

2 marks

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

(b)

Find an expression, in terms of x, for the area of the rectangle. Give your answer without brackets, in its simplest form.

2 marks

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

Hint

Perimeter means adding all four sides; area means length times width.

Worked solution

Here is a rectangle.

All the lengths are in centimetres.

Part (a)

Find an expression, in terms of x, for the perimeter of the rectangle. Give your answer in its simplest form.

  1. Perimeter =(x+5)+(x-1)+(x+5)+(x-1).
  2. x+x+x+x=4x and 5-1+5-1=8.
  3. Perimeter =4x+8 cm.

Part (b)

Find an expression, in terms of x, for the area of the rectangle. Give your answer without brackets, in its simplest form.

  1. Area =(x+5)(x-1).
  2. x\times x=x^2, x\times(-1)=-x, 5\times x=5x and 5\times(-1)=-5.
  3. x^2-x+5x-5=x^2+4x-5 cm².

Question 16Challenge3 marks

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

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

Hint

Expand the double brackets and the single bracket separately, then collect like terms.

Worked solution

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

  1. (x+4)(x+2)=x^2+2x+4x+8=x^2+6x+8.
  2. 3(x-5)=3x-15.
  3. x^2+6x+8+3x-15.
  4. 6x+3x=9x and 8-15=-7, so the answer is x^2+9x-7.

Question 17Challenge2 marks

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

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

Hint

Keep the minus signs with the numbers: -2\times(-4) is positive.

Worked solution

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

  1. 5x\times3x=15x^2, 5x\times(-4)=-20x, -2\times3x=-6x and -2\times(-4)=8.
  2. 15x^2-20x-6x+8.
  3. -20x-6x=-26x, so the answer is 15x^2-26x+8.

Question 18Challenge4 marks

Here are two rectangles, P and Q.

All the lengths are in centimetres, and x is greater than 2

The area of rectangle P is greater than the area of rectangle Q.

Find an expression, in terms of x, for how much greater. Give your answer in its simplest form.

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

Hint

Expand an expression for each area, then subtract the area of Q from the area of P.

Worked solution

Here are two rectangles, P and Q.

All the lengths are in centimetres, and x is greater than 2

The area of rectangle P is greater than the area of rectangle Q.

Find an expression, in terms of x, for how much greater. Give your answer in its simplest form.

  1. Area of P =(x+6)(x+1)=x^2+x+6x+6=x^2+7x+6.
  2. Area of Q =(x+4)(x+2)=x^2+2x+4x+8=x^2+6x+8.
  3. Difference =(x^2+7x+6)-(x^2+6x+8).
  4. x^2-x^2=0, 7x-6x=x and 6-8=-2.
  5. The area of P is x-2 cm² greater.

Question 19Challenge3 marks

(ax+3)(x+2)=4x^2+bx+6

a and b are numbers.

(a)

Work out the value of a.

1 mark

(b)

Work out the value of b.

2 marks

Hint

Expand the left-hand side with a in it, then match the x^2 terms and the x terms.

Worked solution

(ax+3)(x+2)=4x^2+bx+6

a and b are numbers.

Part (a)

Work out the value of a.

  1. ax\times x=ax^2, and this must equal 4x^2.
  2. So a=4.

Part (b)

Work out the value of b.

  1. With a=4: (4x+3)(x+2)=4x^2+8x+3x+6.
  2. 8x+3x=11x, so (4x+3)(x+2)=4x^2+11x+6.
  3. So b=11.

Question 20Challenge3 marks

Expand and simplify (x+5)(x+1)+(x+3)^2

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

Hint

Expand each part separately, remembering that (x+3)^2 means (x+3)(x+3), then collect like terms.

Worked solution

Expand and simplify (x+5)(x+1)+(x+3)^2

  1. (x+5)(x+1)=x^2+x+5x+5=x^2+6x+5.
  2. (x+3)^2=(x+3)(x+3)=x^2+3x+3x+9=x^2+6x+9.
  3. x^2+6x+5+x^2+6x+9.
  4. x^2+x^2=2x^2, 6x+6x=12x and 5+9=14.
  5. The answer is 2x^2+12x+14.

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