1. fACTORISING quadratics

2. What is a quadratic expression?
𝑎 𝑥 2 +𝑏𝑥+𝑐 𝑤ℎ𝑒𝑟𝑒 𝑎,𝑏 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
In a quadratic expression, the HIGHEST POWER of 𝑥 is 2!

3. Discuss with a partner what you think the method could be by observing these answers
fACTORISING
𝒙 +𝟐 𝒙+𝟒 = 𝒙 𝟐 +𝟔𝒙+𝟖 𝒙 +𝟓 𝒙+𝟑 = 𝒙 𝟐 +𝟖𝒙+𝟏𝟓 𝒙 −𝟖 𝒙−𝟒 = 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟐 𝒙 −𝟑 𝒙−𝟑 = 𝒙 𝟐 −𝟔𝒙+𝟗

4. Observations 𝒙 +𝟐 𝒙+𝟒 = 𝒙 𝟐 +𝟔𝒙+𝟖
𝒙 𝟐 opens up as 𝒙 𝒂𝒏𝒅 𝒙
The signs are both plus
= +6
+ 2 x + 4 = +8

5. Observations 𝒙 −𝟖 𝒙−𝟒 = 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟐
𝒙 𝟐 opens up as 𝒙 𝒂𝒏𝒅 𝒙
The signs are both negative inside the brackets but open up as – and +
= - 12
-8 x -4 = + 32

6. METHOD TO Factorise type1 quadratic 𝒙 𝟐 +𝟔𝒙+𝟖= 𝒙 +𝟐 𝒙+𝟒
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Ask the Important Question:
Which TWO numbers give me +8 when I multiply them and +6 when I add them?

7. METHOD TO Factorise type1 quadratic 𝒙 𝟐 +𝟖𝒙+𝟏𝟓= 𝒙 +𝟑 𝒙+𝟓
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Ask the Important Question:
Which TWO numbers give me +15 when I multiply them and +8 when I add them?

8. METHOD TO Factorise type1 quadratic 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟐= 𝒙 −𝟒 𝒙−𝟖
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Ask the Important Question:
Which TWO numbers give me +32 when I multiply them and -12 when I add them?

9. METHOD TO Factorise type1 quadratic 𝒙 𝟐 −𝟔𝒙+𝟗= 𝒙 −𝟑 𝒙−𝟑
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Ask Important Question:
Which TWO numbers give me +9 when I multiply them and -6 when I add them?

10. Type 1 Quadratic: second sign + Bracket signs are THE SAME
What’s the shortcut?
Type 1 Quadratic: second sign +
Bracket signs are THE SAME
𝒙 𝟐 +𝟔𝒙+𝟖= 𝒙 +𝟐 𝒙+𝟒
𝒙 𝟐 +𝟖𝒙+𝟏𝟓= 𝒙 +𝟑 𝒙+𝟓
𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟐= 𝒙 −𝟒 𝒙−𝟖
𝒙 𝟐 −𝟔𝒙+𝟗= 𝒙 −𝟑 𝒙−𝟑

11. Mini-whiteboard Practise !
Any questions?
Mini-whiteboard Practise !

12. Factorise 𝒂 𝟐 +𝟏𝟎𝒂+𝟐𝟒 𝒂+𝟔 𝒂 +𝟒 𝒐𝒓 (𝒂+𝟒)(𝒂+𝟔) Ask Important Question:
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒂 𝟐 as 𝒂 𝒂𝒏𝒅 𝒂, one in each bracket (𝒂 )(𝒂 )
Ask Important Question:
Which TWO numbers give me +24 when I multiply them and +10 when I add them?
6 and 4
𝒂+𝟔 𝒂 +𝟒 𝒐𝒓 (𝒂+𝟒)(𝒂+𝟔)

13. Factorise 𝒑 𝟐 −𝟏𝟔𝒑+𝟔𝟑
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒑 𝟐 as 𝒑 𝒂𝒏𝒅 𝒑, one in each bracket (𝒑 )(𝒑 )
Ask Important Question:
Which TWO numbers give me +63 when I multiply them and −16 when I add them?
If when multiplied we get a positive number and when put together we get a negative number then BOTH numbers should be
NEGATIVE

14. Which TWO numbers give me +63 when I multiply them and −16 when I add them?
Factors of 63 −𝟏 × −𝟔𝟑 −𝟏−𝟔𝟑 = −𝟔𝟒 −𝟑 × −𝟐𝟏 −𝟑−𝟐𝟏 = −𝟐𝟒 −𝟕 × −𝟗 −𝟕−𝟗=−𝟏𝟔 𝒑 𝟐 −𝟏𝟔𝒑+𝟔𝟑=(𝒑−𝟕)(𝒑−𝟗)

15. (𝒙 + 𝟐)(𝒙+𝟏𝟎) Factorise 𝒙 𝟐 +𝟏𝟐𝒙+𝟐𝟎
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a +, then both brackets are +
Important Question:
Which TWO numbers give me +20 when I multiply them and +12 when I add them?
(𝒙 + 𝟐)(𝒙+𝟏𝟎)

16. (𝒙 −𝟏𝟓)(𝒙−𝟐) Factorise 𝒙 𝟐 −𝟏𝟕𝒙+𝟑𝟎
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a −, then both brackets are −
Important Question:
Which TWO numbers give me +30 when I multiply them and -17 when I add them?
(𝒙 −𝟏𝟓)(𝒙−𝟐)

17. (𝒙 +𝟓)(𝒙+𝟔) Factorise 𝒙 𝟐 +𝟏𝟏𝒙+𝟑𝟎
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a +, then both brackets are +
Important Question:
Which TWO numbers give me +30 when I multiply them and +11 when I add them?
(𝒙 +𝟓)(𝒙+𝟔)

18. (𝒙 +𝟔)(𝒙+𝟑) Factorise 𝒙 𝟐 +𝟗𝒙+𝟏𝟖
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a +, then both brackets are +
Important Question:
Which TWO numbers give me +18 when I multiply them and +9 when I add them?
(𝒙 +𝟔)(𝒙+𝟑)

19. (𝒙 −𝟕)(𝒙−𝟔) Factorise 𝒙 𝟐 −𝟏𝟑𝒙+𝟒𝟐
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a −, then both brackets are −
Important Question:
Which TWO numbers give me +42 when I multiply them and -13 when I add them?
(𝒙 −𝟕)(𝒙−𝟔)

20. (𝒙 +𝟏𝟖)(𝒙+𝟏) Factorise 𝒙 𝟐 +𝟏𝟗𝒙+𝟏𝟖
𝑶𝒑𝒆𝒏 𝒖𝒑 𝒙 𝟐 as 𝒙 𝒂𝒏𝒅 𝒙, one in each bracket
Since the 2nd sign is a +, then the signs are the same
Since the 1st sign is also a +, then both brackets are +
Important Question:
Which TWO numbers give me +18 when I multiply them and +19 when I add them?
(𝒙 +𝟏𝟖)(𝒙+𝟏)

21. STARS & WISHES What did you understand well about this lesson?
What do you wish to understand better?

22. Homework factorising type1 quadratic
STP 9 Pg 145 Ex 7b No. 5,11,14 Pg 146 Ex 7c No. 4,7,9

23. 3. 𝑥 2 +7𝑥+12 3 4 𝑥 + 𝑥 + Important Question:
𝑥 2 +7𝑥+12
𝑥 𝑥 +
Important Question:
Which TWO numbers give me +12 when I multiply them and +7 when I add them? 3 4
Factors of 12
1 x 12
2 x 6
3 x 4
= 13
2 + 6 = 8
3 + 4 = 7

24. 5. 𝑥 2 +21𝑥+20 1 𝑥 + 𝑥 + 20 Important Question:
𝑥 2 +21𝑥+20
𝑥 𝑥 +
Important Question:
Which TWO numbers give me +20 when I multiply them and +21 when I add them? 1 20
Factors of 20
1 x 20
2 x 10
4 x 5
= 21
= 12
4 + 5 = 9

25. 11. 𝑥 2 +8𝑥+16 𝑥 + 𝑥 + 4 4 Important Question:
𝑥 2 +8𝑥+16
𝑥 𝑥 +
Important Question:
Which TWO numbers give me +16 when I multiply them and +8 when I add them? 4 4
Factors of 16
1 x 16
2 x 8
4 x 4
= 17
2 + 8 = 10
4 + 4 = 8

26. 14. 𝑥 2 +22𝑥+40 𝑥 + 𝑥 + 2 20 Important Question:
𝑥 2 +22𝑥+40
𝑥 𝑥 +
Important Question:
Which TWO numbers give me +40 when I multiply them and +22 when I add them? 2 20
= 41
= 22
= 14
5 + 8 = 13
Factors of 40
1 x 40
2 x 20
4 x 10
5 x 8

27. 20. 𝑥 2 +14𝑥+40 𝑥 + 𝑥 + 4 10 Important Question:
𝑥 2 +14𝑥+40
𝑥 𝑥 +
Important Question:
Which TWO numbers give me +40 when I multiply them and +14 when I add them? 4 10
= 41
= 22
= 14
5 + 8 = 13
Factors of 40
1 x 40
2 x 20
4 x 10
5 x 8

28. 2. 𝑥 2 −7𝑥+12 𝑥 − 𝑥 − 3 4 Important Question:
𝑥 2 −7𝑥+12
𝑥 − 𝑥 −
Important Question:
Which TWO numbers give me +12 when I multiply them and -7 when I add them? 3 4
Factors of 12
-1 x -12
-2 x -6
-3 x -4
= -13
= -8
= -7

29. 4. 𝑥 2 −11𝑥+28 𝑥 − 𝑥 − 4 7 Important Question:
𝑥 2 −11𝑥+28
𝑥 − 𝑥 −
Important Question:
Which TWO numbers give me +28 when I multiply them and -11 when I add them? 4 7
Factors of 28
-1 x -28
-2 x -14
-4 x -7
= -29
= -16
= -11

30. 7. 𝑥 2 −16𝑥+15 𝑥 − 𝑥 − 1 15 Important Question:
𝑥 2 −16𝑥+15
𝑥 − 𝑥 −
Important Question:
Which TWO numbers give me +15 when I multiply them and -16 when I add them? 1 15
Factors of 15
-1 x -15
-3 x -5
= -16
= -8

31. 9. 𝑥 2 −18𝑥+32 𝑥 − 𝑥 − 2 16 Important Question:
𝑥 2 −18𝑥+32
𝑥 − 𝑥 −
Important Question:
Which TWO numbers give me +32 when I multiply them and -18 when I add them? 2 16
Factors of 32
-1 x -32
-2 x -16
-4 x -8
= -33
= -18
= -12

32. 10. 𝑥 2 −16𝑥+63 𝑥 − 𝑥 − 7 9 Important Question:
𝑥 2 −16𝑥+63
𝑥 − 𝑥 −
Important Question:
Which TWO numbers give me +63 when I multiply them and -16 when I add them? 7 9
Factors of 63
-1 x -63
-3 x -21
-7 x -9
= -64
= -24
= -16