fACTORISING quadratics

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!

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

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

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 + -8 - 4 = - 12 -8 x -4 = + 32

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?

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?

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?

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?

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

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

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)

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

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)

(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)

(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)

(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)

(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)

(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)

(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)

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

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

3. 𝑥 2 +7𝑥+12 3 4 𝑥 + 𝑥 + Important Question: 3. 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 1 + 12 = 13 2 + 6 = 8 3 + 4 = 7

5. 𝑥 2 +21𝑥+20 1 𝑥 + 𝑥 + 20 Important Question: 5. 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 1 + 20 = 21 2 + 10 = 12 4 + 5 = 9

11. 𝑥 2 +8𝑥+16 𝑥 + 𝑥 + 4 4 Important Question: 11. 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 1 + 16 = 17 2 + 8 = 10 4 + 4 = 8

14. 𝑥 2 +22𝑥+40 𝑥 + 𝑥 + 2 20 Important Question: 14. 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 1 + 40 = 41 2 + 20 = 22 4 + 10 = 14 5 + 8 = 13 Factors of 40 1 x 40 2 x 20 4 x 10 5 x 8

20. 𝑥 2 +14𝑥+40 𝑥 + 𝑥 + 4 10 Important Question: 20. 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 1 + 40 = 41 2 + 20 = 22 4 + 10 = 14 5 + 8 = 13 Factors of 40 1 x 40 2 x 20 4 x 10 5 x 8

2. 𝑥 2 −7𝑥+12 𝑥 − 𝑥 − 3 4 Important Question: 2. 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 -1 - 12 = -13 -2 - 6 = -8 -3 - 4 = -7

4. 𝑥 2 −11𝑥+28 𝑥 − 𝑥 − 4 7 Important Question: 4. 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 -1 - 28 = -29 -2 - 14 = -16 -4 - 7 = -11

7. 𝑥 2 −16𝑥+15 𝑥 − 𝑥 − 1 15 Important Question: 7. 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 -1 - 15 = -16 -3 - 5 = -8

9. 𝑥 2 −18𝑥+32 𝑥 − 𝑥 − 2 16 Important Question: 9. 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 -1 - 32 = -33 -2 - 16 = -18 -4 - 8 = -12

10. 𝑥 2 −16𝑥+63 𝑥 − 𝑥 − 7 9 Important Question: 10. 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 -1 - 63 = -64 -3 - 21 = -24 -7 - 9 = -16