Expanding brackets and factorising

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Notes de leçon

Expanding a Single Bracket

To expand a bracket, multiply the term outside by each term inside.
Example: 3(x + 4) = 3x + 12.
Example: 5(2x − 3) = 10x − 15.
Remember to multiply both the number and the variable parts.
If the term outside is negative, the signs inside the bracket change: −2(x + 5) = −2x − 10.

Expanding and Simplifying Two Brackets

To expand (x + a)(x + b), multiply each term in the first bracket by each term in the second.
This gives four products: x·x + x·b + a·x + a·b.
Simplify by collecting like terms: x² + (a + b)x + ab.
Example: (x + 2)(x + 3) = x² + 5x + 6.
Example: (x − 4)(x + 1) = x² − 3x − 4.

Squared Brackets

A squared bracket like (x + a)² means (x + a)(x + a).
Expand: (x + a)² = x² + 2ax + a².
Example: (x + 5)² = x² + 10x + 25.
Example: (x − 3)² = x² − 6x + 9.
Be careful: (x + a)² is not x² + a²; the middle term 2ax is essential.

Factorising by Taking Out a Common Factor

Factorising is the reverse of expanding: write an expression as a product of factors.
Find the highest common factor (HCF) of all terms.
Write the HCF outside a bracket and divide each term by it inside.
Example: 6x + 9 = 3(2x + 3).
Example: 4x² − 8x = 4x(x − 2).
Always check by expanding your answer to see if you get the original expression.

Factorising Quadratic Expressions x^2 + bx + c

Look for two numbers that multiply to give c and add to give b.
Then write the expression as (x + p)(x + q) where p and q are those numbers.
Example: x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7.
Example: x² − 5x + 6 = (x − 2)(x − 3) because (−2) × (−3) = 6 and (−2) + (−3) = −5.
If c is negative, the two numbers have opposite signs.
If c is positive and b is negative, both numbers are negative.

Difference of Two Squares

The difference of two squares is an expression of the form a² − b².
It factorises as (a + b)(a − b).
Example: x² − 9 = (x + 3)(x − 3).
Example: 4x² − 25 = (2x + 5)(2x − 5).
This pattern works because (a + b)(a − b) = a² − b² when expanded.

Checking Your Work

After factorising, expand your answer to check it matches the original expression.
After expanding, simplify by collecting like terms.
Use substitution: pick a value for x and check both expressions give the same result.
Common mistake: forgetting to multiply all terms inside the bracket.
Common mistake: incorrect signs when expanding or factorising.

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1.Expand 3(x + 4).
Easy
- A3x + 12
- B3x + 4
- Cx + 12
- D3x − 12
2.Expand and simplify (x + 2)(x + 5).
Medium
- Ax² + 7x + 10
- Bx² + 10x + 7
- Cx² + 7x + 7
- Dx² + 10
3.Factorise fully 6x + 9.
Medium
- A3(2x + 3)
- B3(2x + 9)
- C6(x + 3)
- D9(x + 1)
4.Factorise x² + 7x + 12.
Medium
- A(x + 3)(x + 4)
- B(x + 2)(x + 6)
- C(x + 1)(x + 12)
- D(x − 3)(x − 4)
5.Factorise x² − 9.
Medium
- A(x − 3)(x + 3)
- B(x − 9)(x + 1)
- C(x − 3)²
- D(x − 3)(x − 3)
6.Which of the following are correct factorisations? (select all that apply)
Medium
- Ax² − 16 = (x − 4)(x + 4)
- Bx² + 6x + 8 = (x + 2)(x + 4)
- C2x + 6 = 2(x + 6)
- Dx² − 5x + 6 = (x − 2)(x − 3)
- Ex² + 4 = (x + 2)(x + 2)
7.The expression x² − 25 can be factorised as (x − 5)(x + 5).
Easy
True or false?
8.Match each expression with its fully factorised form.
Medium
- 4x + 8
- x² − 1
- x² + 5x + 6
- (x + 2)(x + 3)
- 4(x + 2)
- (x − 1)(x + 1)

Plus de questions

9.Put the steps for expanding and simplifying (x + 3)(x + 4) in the correct order.
Medium
- Write x² + 4x + 3x + 12
- Multiply the terms: x × x, x × 4, 3 × x, 3 × 4
- Combine like terms to get x² + 7x + 12
- Identify the two binomials to multiply
10.Write an expression for the area of the rectangle in expanded form.
Medium
- Ax² + 7x + 10
- Bx² + 10x + 7
- Cx² + 10
- D2x + 7
11.Which of the following expressions are fully factorised? (select all that apply)
Medium
- A3(x + 2)
- B2(3x + 1)
- Cx(2x + 3)
- D6x + 9
- Ex² + 5x + 6
12.Put the steps for factorising x² + 9x + 20 into the correct order.
Medium
- Find two numbers that multiply to 20 and add to 9.
- The numbers are 4 and 5.
- Write the factorised form as (x + 4)(x + 5).
- Check by expanding: (x + 4)(x + 5) = x² + 9x + 20.
13.The expression x² + 6x + 9 can be written as (x + 3)².
Easy
True or false?
14.Which expression is the fully factorised form of 12x² − 18x?
Medium
- A6x(2x − 3)
- B6(2x² − 3x)
- C3x(4x − 6)
- D2x(6x − 9)
15.Which expression is the fully factorised form of x² − 49?
Medium
- A(x − 7)(x + 7)
- B(x − 7)²
- C(x + 7)²
- D(x − 49)(x + 1)
16.Which of the following statements about factorising are true? (select all that apply)
Hard
- AFactorising is the inverse process of expanding brackets.
- BEvery expression with a common factor can be factorised by taking out that factor.
- Cx² − 9 can be factorised as (x − 3)(x + 3).
- Dx² + 4 cannot be factorised using integers.
- Ex² + 4x + 4 factorises as (x + 2)(x − 2).
17.Expand 4(3x + 2).
Easy
- A12x + 8
- B12x + 2
- C7x + 8
- D12x − 8
18.Expand and simplify (x + 4)(x − 2).
Medium
- Ax² + 2x − 8
- Bx² − 2x − 8
- Cx² + 2x + 8
- Dx² − 6x − 8
19.Factorise fully 8x + 12.
Medium
- A4(2x + 3)
- B2(4x + 6)
- C4(2x + 12)
- D8(x + 12)
20.Factorise x² + 5x + 6.
Medium
- A(x + 2)(x + 3)
- B(x + 1)(x + 6)
- C(x − 2)(x − 3)
- D(x + 5)(x + 1)
21.Factorise x² − 16.
Medium
- A(x − 4)(x + 4)
- B(x − 4)²
- C(x + 4)²
- D(x − 8)(x + 2)
22.Which of the following expressions are equivalent to (x + 3)(x + 3)? (select all that apply)
Medium
- Ax² + 6x + 9
- B(x + 3)²
- Cx² + 9
- Dx² + 3x + 9
23.The expression x² − 4 can be factorised as (x − 2)(x + 2).
Easy
True or false?
24.Put the steps for expanding and simplifying (x + 2)(x + 5) in the correct order.
Medium
- Write x² + 5x + 2x + 10
- Multiply each term in the first bracket by each term in the second
- Collect like terms to get x² + 7x + 10
25.A rectangle has length (x + 7) cm and width (x − 2) cm. Write an expression for its area in expanded form.
Medium
- Ax² + 5x − 14
- Bx² − 5x − 14
- Cx² + 9x − 14
- Dx² + 5x + 14
26.A square has side length (2x + 3) cm. Which expression gives its area in expanded form?
Medium
- A4x² + 12x + 9
- B2x² + 12x + 9
- C4x² + 6x + 9
- D4x² + 12x + 6
27.The area of a rectangle is (x² + 7x + 12) cm². Its length is (x + 4) cm. Which expression gives its width?
Hard
- Ax + 3
- Bx + 4
- Cx + 8
- Dx + 7
28.A triangle has base (x + 6) cm and height (x + 2) cm. Which expression gives its area in expanded form?
Medium
- A0.5x² + 4x + 6
- Bx² + 8x + 12
- C0.5x² + 8x + 12
- Dx² + 4x + 6
29.The expression x² − 64 can be factorised as (x − 8)(x + 8).
Medium
True or false?
30.A square has area (x² + 10x + 25) cm². Which expression gives the side length of the square?
Medium
- Ax + 5
- Bx + 10
- Cx + 25
- Dx + 2
31.A rectangle has length (x + 6) cm and width (x − 2) cm. Which of the following statements are true? (select all that apply)
Hard
- AThe area is x² + 4x − 12 cm².
- BThe perimeter is 4x + 8 cm.
- CThe area is x² + 4x + 12 cm².
- DThe perimeter is 2x + 4 cm.
- EThe area when x = 4 is 20 cm².
32.What does it mean to factorise an expression?
Easy
- AWrite it as a product of factors
- BWrite it as a sum of terms
- CMultiply out the brackets
- DFind the value of the expression
33.Which of these is a factorised form?
Easy
- A(x + 2)(x - 3)
- Bx² - x - 6
- Cx² + 5x + 6
- Dx(x + 5) + 6
34.Expanding brackets and factorising are inverse processes.
Easy
True or false?
35.Which expression is the fully factorised form of 10x + 15?
Easy
- A5(2x + 3)
- B5(2x + 15)
- C10(x + 5)
- D5(2x + 5)
36.Which of the following are factors of 12x²? (select all that apply)
Medium
- A3x
- B4x
- C2x²
- D6x
- Ex³
37.Match each expression with its factorised form.
Medium
- x² + 8x + 15
- x² - 8x + 15
- x² - 15x + 56
- x² + 15x + 56
- (x + 3)(x + 5)
- (x - 7)(x - 8)
38.Which expression is the fully factorised form of 4x² - 25?
Medium
- A(2x - 5)(2x + 5)
- B(2x - 5)²
- C(4x - 5)(x + 5)
- D(2x - 25)(2x + 1)
39.Put the steps for factorising x² + 11x + 24 in the correct order.
Medium
- Find two numbers that multiply to 24 and add to 11.
- The numbers are 3 and 8.
- Write the factorised form as (x + 3)(x + 8).
40.Put the steps for factorising x² + 9x + 20 in the correct order.
Medium
- Find two numbers that multiply to 20 and add to 9.
- The numbers are 4 and 5 because 4 × 5 = 20 and 4 + 5 = 9.
- Write the factorised form as (x + 4)(x + 5).
41.Expand 5(2x + 3). What is the coefficient of x in the expanded expression?
Medium
42.Complete the factorisation by filling in the missing numbers.
Medium
x² + 9x + 20 = (x + ____)(x + ____)
43.If x = 3, what is the area of the rectangle in cm²?
Medium
44.Complete the factorisation of x² + 8x + 15 by filling in the missing numbers.
Medium
x² + 8x + 15 = (x + ____)(x + ____)
45.Expand and simplify (x + 6)(x − 1). What is the coefficient of x in the simplified expression?
Medium
46.What is the highest common factor of 8x² and 12x?
Easy
47.Expand 7(2x + 1). What is the coefficient of x in the expanded expression?
Easy
48.Complete the factorisation by filling in the missing number.
Medium
5x + 15 = 5(x + ____)
49.Complete the factorisation of x² + 7x + 6 by filling in the missing numbers.
Medium
x² + 7x + 6 = (x + ____)(x + ____)
50.Expand 6(2x + 5). What is the coefficient of x in the expanded expression?
Easy
51.Expand and simplify (x + 7)(x − 3). What is the coefficient of x in the simplified expression?
Medium
52.Complete the factorisation of x² + 10x + 21 by filling in the missing numbers.
Easy
x² + 10x + 21 = (x + ____)(x + ____)
53.Explain how to factorise a quadratic expression of the form x² + bx + c.
Medium
54.Explain how to factorise a quadratic expression of the form x² + bx + c, where b and c are integers.
Medium
55.Explain how to expand and simplify the product of two brackets, such as (x + 3)(x + 4).
Hard
56.A rectangle has length (x + 5) cm and width (x − 2) cm. The area is 18 cm². Form an equation and solve it to find the positive value of x.
Hard
57.Expand 5(2x + 3). What is the coefficient of x in the expanded expression?
Medium
- A11
- B10
- C9
- D12
58.If x = 3, what is the area of the rectangle in cm²?
Medium
- A44 cm²
- B36 cm²
- C48 cm²
- D40 cm²
59.Expand and simplify (x + 6)(x − 1). What is the coefficient of x in the simplified expression?
Medium
- A6
- B4
- C5
- D7
60.Expand 7(2x + 1). What is the coefficient of x in the expanded expression?
Easy
- A15
- B14
- C13
- D16
61.Expand 6(2x + 5). What is the coefficient of x in the expanded expression?
Easy
- A14
- B11
- C13
- D12
62.Expand and simplify (x + 7)(x − 3). What is the coefficient of x in the simplified expression?
Medium
- A4
- B5
- C3
- D6

Notes de révision

## Expanding Brackets and Factorising ### Expanding Brackets - **Single term over a bracket:** Multiply the term outside by each term inside. - Example: \(3(x + 4) = 3x + 12\) - **Two brackets:** Use the distributive law (often called FOIL). - For \((x + a)(x + b)\), expand to get \(x^2 + (a+b)x + ab\). - Example: \((x + 2)(x + 3) = x^2 + 5x + 6\) - **Squared brackets:** \((x + a)^2 = (x + a)(x + a) = x^2 + 2ax + a^2\). - Example: \((x + 5)^2 = x^2 + 10x + 25\) - **Simplifying:** After expanding, collect like terms. ### Factorising - **Common factor:** Take out the highest common factor (HCF) from all terms. - Example: \(6x + 9 = 3(2x + 3)\) - **Quadratic expressions \(x^2 + bx + c\):** Find two numbers that multiply to \(c\) and add to \(b\). - Then write as \((x + p)(x + q)\) where \(p \times q = c\) and \(p + q = b\). - Example: \(x^2 + 7x + 12 = (x + 3)(x + 4)\) because \(3 \times 4 = 12\) and \(3 + 4 = 7\). - **Difference of two squares:** \(a^2 - b^2 = (a - b)(a + b)\). - Example: \(x^2 - 9 = (x - 3)(x + 3)\) ### Key Points - Factorising is the reverse of expanding. - Always check for a common factor first. - For quadratics, ensure the expression is in the form \(x^2 + bx + c\) (coefficient of \(x^2\) is 1). - Practise recognising patterns like the difference of two squares. ### Common Mistakes - Forgetting to multiply all terms inside the bracket. - Losing negative signs when expanding. - Not fully factorising (e.g., leaving a common factor). - Mixing up the sum and product when factorising quadratics.

Expanding: Multiplying out brackets to remove them and simplify the expression.
Factorising: Writing an expression as a product of its factors (the reverse of expanding).
Common factor: A number or variable that divides exactly into all terms of an expression.
Quadratic expression: An expression of the form ax^2 + bx + c, where a, b, and c are numbers and a ≠ 0.
Difference of two squares: An expression of the form a^2 - b^2, which factorises as (a - b)(a + b).

Cartes mémo

Expand: 3(x + 4)

Multiply each term inside the bracket by 3: 3x + 12.

Expand: 5(2x − 3)

Multiply each term by 5: 10x − 15.

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

Expand each bracket: 2x + 6 + 4x + 4. Then collect like terms: 6x + 10.

Expand: (x + 2)(x + 5)

Use FOIL: x² + 5x + 2x + 10 = x² + 7x + 10.

Expand: (x − 3)(x + 4)

x² + 4x − 3x − 12 = x² + x − 12.

Expand: (x + 6)(x − 2)

x² − 2x + 6x − 12 = x² + 4x − 12.

Expand: (x − 5)(x − 3)

x² − 3x − 5x + 15 = x² − 8x + 15.

Expand: (x + 7)²

Write as (x + 7)(x + 7) and expand: x² + 7x + 7x + 49 = x² + 14x + 49.

Expand: (x − 4)²

(x − 4)(x − 4) = x² − 4x − 4x + 16 = x² − 8x + 16.

Factorise fully: 6x + 9

The highest common factor of 6x and 9 is 3, so 6x + 9 = 3(2x + 3).

Factorise fully: 12x² − 18x

The highest common factor is 6x, so 12x² − 18x = 6x(2x − 3).

Factorise: x² + 7x + 12

Find two numbers that multiply to 12 and add to 7: 3 and 4. So x² + 7x + 12 = (x + 3)(x + 4).

Factorise: x² − 5x + 6

Two numbers that multiply to 6 and add to −5: −2 and −3. So x² − 5x + 6 = (x − 2)(x − 3).

Factorise: x² + 2x − 15

Two numbers that multiply to −15 and add to 2: 5 and −3. So x² + 2x − 15 = (x + 5)(x − 3).

Factorise: x² − 9

This is the difference of two squares: x² − 9 = (x + 3)(x − 3).

Factorise: x² − 25

Difference of two squares: x² − 25 = (x + 5)(x − 5).

What is the difference of two squares pattern?

a² − b² = (a + b)(a − b). For example, x² − 16 = (x + 4)(x − 4).

How do you expand a single term over a bracket?

Multiply the term outside the bracket by each term inside the bracket. For example, 4(2x + 3) = 8x + 12.

What does it mean to factorise an expression?

Write it as a product of factors. For example, factorising x² + 5x + 6 gives (x + 2)(x + 3).

How do you factorise a quadratic of the form x² + bx + c?

Find two numbers that multiply to c and add to b, then write (x + first number)(x + second number).

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

Expand each product: x² + 3x + 2 + x² + 7x + 12. Collect like terms: 2x² + 10x + 14.

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