KS3 Maths · Year 8 · Chapter 8: Algebra · Lesson 1

Expanding the product of two brackets

Year 8About 70 minutesNo calculator49 marks

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Date

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Quick questions to start.

A1
Needed today
Expand $4 (3 x - 2)$ .
[1 mark]
Answer
A2
Needed today
Work out $- 6 \times - 9$ .
[1 mark]
Answer
A3
Last lesson
Write $\frac{x}{3} - \frac{x}{5}$ as a single fraction.
[1 mark]
Answer
A4
From chapter 2
Expand $x (2 x - 7)$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Expand and simplify $(x + 5) (x + 7)$ .

Not drawn accurately

Hint Put one bracket along the top of a grid and the other down the side. Multiply to fill each box, then add the four boxes.

B1
Expand and simplify $(y + 7) (y + 8)$ .
[2 marks]
Answer

Example 2

Expand and simplify $(x - 2) (x + 8)$ .

Hint Keep the sign with its number: the boxes are

x^{2}

8 x

- 2 x

and

- 2 \times 8 = - 16

B2
Expand and simplify $(t + 1) (t - 8)$ .
[2 marks]
Answer

Example 3

Expand and simplify $(x - 8)^{2}$ . Explain why the answer is not $x^{2} + 64$ .

Hint

(x - 8)^{2}

means

(x - 8) (x - 8)

. Fill in all four boxes.

B3
Expand and simplify $(n + 12)^{2}$ .
[2 marks]
Answer

Example 4

Expand and simplify $(3 x + 4) (2 x - 7)$ .

Hint

3 x \times 2 x = 6 x^{2}

. Then

3 x \times - 7

4 \times 2 x

and

4 \times - 7

B4
Expand and simplify $(4 x + 1) (x - 3)$ .
[2 marks]
Answer

Example 5

Expand and simplify $(2 x + y) (x - 3 y)$ .

Hint The boxes are

2 x^{2}

- 6 x y

x y

and

- 3 y^{2}

- 6 x y

and

x y

are like terms.

B5
Expand and simplify $(x + 4 y) (3 x - y)$ .
[3 marks]
Answer

Practice

Year 8

C1
The rectangle is split into four parts. Write down the area of each part. Then write the area of the whole rectangle, $(x + 11) (x + 2)$ , as simply as possible.
[3 marks]
Not drawn accurately
Answer
C2
Expand and simplify $(5 - x) (2 + x)$ .
[2 marks]
Answer

C3
Work out the missing number: $(x - 9) (x - □) = x^{2} - 11 x + 18$ .
[2 marks]
Answer
C4
Expand and simplify $(x + 5)^{2} - (x + 1) (x + 9)$ .
[3 marks]
Answer
C5
Show that $(3 x + 6)^{2}$ is the same as $9 (x + 2)^{2}$ .
[2 marks]

Exam-style questions

GCSE Foundation

D1
Ella says: " $(x + 14)^{2} = x^{2} + 196$ ."
Ella is wrong. Explain her mistake and expand $(x + 14)^{2}$ correctly.
[3 marks]
Answer $(x + 14)^{2} = \dots\dots$

D2
A square of side $x$ cm is cut from the corner of a rectangle $(2 x + 5)$ cm long and $(x + 3)$ cm wide. The shaded part is left.
Not drawn accurately
(a)
Show that the shaded area is $(x^{2} + 11 x + 15)$ cm².
[3 marks]
(b)
Work out the shaded area when $x = 4$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
Solve $(x + 7) (x + 10) = x^{2} + 104$ .
[3 marks]
Answer
D4
A rectangle is $(x + 8)$ cm long and $(x - 4)$ cm wide. A square has sides of $x$ cm. The area of the rectangle is $16$ cm² more than the area of the square. Work out the value of $x$ .
[4 marks]
Not drawn accurately
Answer
D5
Here is an expression: $(2 x - 7)^{2}$ .
(a)
Expand and simplify $(2 x - 7)^{2}$ .
[2 marks]
Answer
(b)
Use your answer with $x = 10$ to work out $1 3^{2}$ . Show your working.
[1 mark]
Answer
Total for D5: 3 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$(x + a) (x + b) = x^{2} + 9 x + c$ , where $a$ and $b$ are positive whole numbers. Work out the largest possible value of $c$ and the smallest possible value of $c$ .
[4 marks]
Hint 1 · What to try
Expand $(x + a) (x + b)$ . The $x$ term is $(a + b) x$ and the number term is $ab$ .
Hint 2 · The first line
So $a + b = 9$ . List the pairs: $1$ and $8$ , $2$ and $7$ , $3$ and $6$ , $4$ and $5$ .
Hint 3 · The full method
Work out $ab$ for each pair.
AnswerLargest $\dots\dots$ smallest $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand two brackets with a grid and collect like terms.
I can square a bracket without leaving out the middle terms.
I can expand brackets with an $x$ coefficient or two letters.
I can use expanding to solve equations and find areas.

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