GCSE Maths · Higher · Chapter 8: Algebraic manipulation · Lesson 2

Expanding brackets and squares

Grades 4–7About 50 minutesNo calculator42 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Expand $6 (5 x - 8)$ .
[1 mark]
Answer
A2
Last lesson
Factorise fully $4 x^{2} + 10 x$ .
[1 mark]
Answer
A3
From chapter 3
What type of correlation would you expect between age of a car and its value?
[1 mark]
Answer
A4
From chapter 6
The bearing of B from A is $32 3^{\circ}$ . Find the bearing of A from B.
[1 mark]
Answer

Grades 4–6

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

Example 1

Expand and simplify $(x + 6) (x - 9)$ .

Hint There are four products. Watch the sign of each one.

Example 2

Expand and simplify $(3 x - 5)^{2}$ .

Hint Write it as

(3 x - 5) (3 x - 5)

. A square has a middle term.

Grades 5–6

C3
$(x + a)^{2} = x^{2} + 18 x + b$ for every value of $x$ . Work out $a$ and $b$ .
[2 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
C4
Expand and simplify $(5 x - 4) (5 x + 4)$ .
[2 marks]
Answer
C5
Expand and simplify $(3 x - 2 y) (x + 4 y)$ .
[2 marks]
Answer
C6
Expand and simplify $(x + 3) (x - 5) - (x - 2)^{2}$ .
[3 marks]
Answer
C7
Expand and simplify $(x - 3)^{3}$ .
[3 marks]
Answer

Grades 6–7

D1
Leo says, " $(x - 5)^{2} = x^{2} + 25$ ."
Explain why Leo is wrong, and expand $(x - 5)^{2}$ correctly.
[2 marks]

D2
A photo is $(x + 6)$ cm wide and $(x + 2)$ cm high. It sits in a frame that is 2 cm wide all the way round. The diagram is not drawn accurately; lengths are in cm.
(a)
Show that the area of the frame alone is $(8 x + 48)$ cm².
[4 marks]
(b)
The frame has an area of 104 cm². Work out the value of $x$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
A cuboid measures $(x + 3)$ cm by $(x + 1)$ cm by $(2 x - 1)$ cm. The diagram is not drawn accurately. Show that its volume is $(2 x^{3} + 7 x^{2} + 2 x - 3)$ cm³.
[3 marks]
D4
$n$ is a whole number. Show that $(n + 7)^{2} - (n - 1)^{2}$ is always a multiple of 16.
[3 marks]

Grade 8 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can expand two brackets, including when a number is negative.
I can square a bracket and remember its middle term.
I can expand three brackets, two at a time.

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