1. Today I will learn to factorise both
linear and quadratic equations.
Grade C
Challenge Objective: To factorise
quadratics that are in the form
ax² + bx + c Grade B

2. Multiplying out simple equations
1. 5x(3y – 4x) You have 8
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

3. Multiplying out simple equations
1. 5x(3y – 4x) You have 7
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

4. Multiplying out simple equations
1. 5x(3y – 4x) You have 6
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

5. Multiplying out simple equations
1. 5x(3y – 4x) You have 5
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

6. Multiplying out simple equations
1. 5x(3y – 4x) You have 4
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

7. Multiplying out simple equations
1. 5x(3y – 4x) You have 3
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

8. Multiplying out simple equations
1. 5x(3y – 4x) You have 2
2. 7(a – 3b) minutes
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

9. Multiplying out simple equations
1. 5x(3y – 4x) You have 1
2. 7(a – 3b) minute
3. 9(ef + 3)
4. 7ab(2a – 5b)
5. 20pq(a – p + q)
6. 15mn(3 – 4m + j)
7. 6dst(5s – 3t + 2d)
8. ab²(c – a + 2b)

10. Answers
1. 5x(3y – 4x) = 15xy – 20x²
2. 7(a – 3b) = 7a – 21b
3. 9(ef + 3) = 9ef + 27
4. 7ab(2a – 5b) = 14a²b – 35ab²
5. 20pq(a – p + q) = 20apq – 20p²q + 20pq²
6. 15mn(3 – 4m + j) = 45mn – 60m²n + 15jmn
7. 6dst(5s – 3t + 2d) = 30ds²t – 18dst² + 12d²st
8. ab²(c – a + 2b) = ab²c – a²b² + 2ab³

11. Multiplying out quadratic equations
• Example 2. (w – 3)(w + 6)

12. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 7
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

13. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 6
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

14. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 5
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

15. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 4
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

16. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 3
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

17. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 2
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

18. Multiplying out quadratic equations
1. (a + 3)(a + 4) You have 1
2. (b – 2)(b + 3) minutes
3. (c – 5)(c – 2)
4. (d + 2)(d – 4)
5. (e – 6)(e + 6)
6. (f + 3)(f – 3)
7. What do you notice about question 5
and 6?

19. Answers
1. (a + 3)(a + 4) = a² + 7a + 12
2. (b – 2)(b + 3) = b² + b - 6
3. (c – 5)(c – 2) = c² - 7c + 10
4. (d + 2)(d – 4) = d² - 2d - 8
5. (e – 6)(e + 6) = e² - 36
6. (f + 3)(f – 3) = f² - 9
7. What do you notice about question 5
and 6?

20. Today I will learn to factorise both
linear and quadratic equations.
Grade C
Challenge Objective: To factorise
quadratics that are in the form
ax² + bx + c Grade B

21. Factorise simple expressions
• Example 3. 5x² + 10x Example 4. 15ab² - 5a + 10b

22. Factorise equations
1. 5ab – 10b You have 8
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

23. Factorise equations
1. 5ab – 10b You have 7
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

24. Factorise equations
1. 5ab – 10b You have 6
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

25. Factorise equations
1. 5ab – 10b You have 5
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

26. Factorise equations
1. 5ab – 10b You have 4
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

27. Factorise equations
1. 5ab – 10b You have 3
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

28. Factorise equations
1. 5ab – 10b You have 2
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

29. Factorise equations
1. 5ab – 10b You have 1
2. 18abc + 6ab – 3abc minutes
3. 20st – 14su
4. 8n²m – 2nm² + 4mn
5. 9pq² + 18pq – 3p²q
6. 36yz + 18z – 12z²
7. 42a³b² – 84a²b
8. 45gh + 63hi – 27h³

30. Answers
1. 5ab – 10b = 5b(a – 2)
2. 18abc + 6ab – 3abc = 3ab(6c + 2 – c)
3. 20st – 14su = 2s(10t – 7u)
4. 8mn² – 2m²n + 4mn = 2mn(4n – m + 2)
5. 9pq² + 18pq – 3p²q = 3pq(3q + 6 – p)
6. 36yz + 18z – 12z² = 6z(6y + 3 – 2z)
7. 42a³b² – 84a²b = 42a²b(ab – 2a)
8. 45gh + 63hi – 27h³ = 9h(5g + 7i – 3h²)

31. Things to remember
Example 5. Factorise Factorise means put in
a² - 8a + 16 brackets.
• x²+ ax + b = (x + ?)(x + ?)
Since everything is
positive.
• x² – ax + b = (x – ?)(x – ?)
Since –ve × –ve = +ve.
• x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
• x²– ax – b = (x + ?)(x – ?)

32. You have 12
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

33. You have 11
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

34. You have 10
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

35. You have 9
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

36. You have 8
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

37. You have 7
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

38. You have 6
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

39. You have 5
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

40. You have 4
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

41. You have 3
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

42. You have 2
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

43. You have 1
minutes Factorise equations
1. a² + 5a + 6 Factorise means put in
2. b² + 7b +12 brackets.
• x²+ ax + b = (x + ?)(x + ?)
3. c² - 6c + 9 Since everything is
4. d² + d – 20 positive.
• x² – ax + b = (x – ?)(x – ?)
5. e² - 12e + 35 Since –ve × –ve = +ve.
6. f² +2f – 8 • x²+ ax – b = (x + ?)(x – ?)
Since +ve × –ve = –ve.
7. g² +2g – 24
• x²– ax – b = (x + ?)(x – ?)
8. h² - 11h + 30

44. Answers
1. a² + 5a + 6 = (a + 2)(a + 3)
2. b² + 7b +12 = (b + 4)(b + 3)
3. c² - 6c + 9 = (c – 3)(c – 3)
4. d² + d – 20 = (d – 4)(d + 5)
5. e² - 12e + 35 = (e – 5)(e – 7)
6. f² +2f – 8 = (f – 2)(f + 4)
7. g² +2g – 24 = (g + 6)(g – 4)
8. h² - 11h + 30 = (h – 5)(h – 6)

45. Plenary
• Today’s learning objective was ‘Today I will
multiply out and factorise both linear and
quadratic equations’ complete the questions
below to prove you have met the objective!
Multiply out
a) 5xy(2x – 4) b) (x – 4)(x + 2)
Factorise
c) 4x² - 12xy d) x² - 3x - 10