Next week is revision week What topics or objectives would you like to revise?

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!

Expanding QUADRATIC EXPRESSIONS Expand the following: (𝒙 +𝟒)(𝒙+𝟐) 𝒙 +𝟓 𝒙+𝟑 (𝒙 −𝟖)(𝒙−𝟒) (𝒙 −𝟑)(𝒙−𝟑)

Expanding QUADRATIC EXPRESSIONS 𝒙 𝟐 +𝟔𝒙+𝟖 𝒙 +𝟒 𝒙 𝟐 +𝟒𝒙 𝒙 (𝒙 +𝟒)(𝒙 +𝟐) x +𝟐𝒙 +𝟖 +𝟐 𝒙 𝟐 +𝟐𝒙 +𝟒𝒙 + 𝟖

Expanding QUADRATIC EXPRESSIONS 𝒙 𝟐 +𝟖𝒙+𝟏𝟓 𝒙 +𝟓 𝒙 𝟐 +𝟓𝒙 𝒙 (𝒙 +𝟓)(𝒙 +𝟑) x +𝟑𝒙 +𝟏𝟓 +𝟑 𝒙 𝟐 +𝟑𝒙 +𝟓𝒙 + 𝟏𝟓

Expanding QUADRATIC EXPRESSIONS 𝒙 𝟐 −𝟏𝟐𝒙+𝟑𝟐 𝒙 −𝟖 𝒙 𝟐 −𝟖𝒙 𝒙 (𝒙 −𝟖)(𝒙 −𝟒) x −𝟒𝒙 +𝟑𝟐 −𝟒 𝒙 𝟐 −𝟒𝒙 −𝟖𝒙 + 𝟑𝟐

Expanding QUADRATIC EXPRESSIONS 𝒙 𝟐 −𝟔𝒙+𝟗 𝒙 −𝟑 𝒙 𝟐 −𝟑𝒙 𝒙 (𝒙 −𝟑)(𝒙 −𝟑) x −𝟑𝒙 +𝟗 −𝟑 𝒙 𝟐 −𝟑𝒙 −𝟑𝒙 + 𝟗

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

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)

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