Factorising Quadratics

Factorising Quadratics Index 1. What are quadratics? 2. Factorising quadratics (coefficient of x 2 is 1) 3. Predicting the signs of the final answer. 4. Factorising Quadratics (coefficient of x 2 is not 1)

Factorising Quadratics What are they? ,[object Object]

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x + x ×4

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x + 3 ×x + x ×4

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x + 3 ×x + x ×4 + 3×4

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x + x ×4 + 3×4

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x add the like terms = x 2 + 7x + 12 + x ×4 + 3×4

Factorising Quadratics What are they? ,[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x add the like terms = x 2 + 7x + 12 + x ×4 + 3×4 In this example the resulting equation is called a QUADRATIC EQUATION. The biggest power of x is 2 in Quadratic Equation. Examples of quadratic equations: x 2 + x + 2 x 2 – 2x + 6 4x 2 – 100 2x 2 – 14x + 20

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],[object Object],[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x = x 2 + 7x + 12 + x ×4 + 3×4

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],[object Object],[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x = x 2 + 7x + 12 + x ×4 + 3×4 Where does the 3 rd term come from?

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],[object Object],[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x = x 2 + 7x + 12 + x ×4 + 3×4 Where does the 3 rd term come from? multiply the last terms of each bracket

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],[object Object],[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x = x 2 + 7x + 12 + x ×4 + 3×4 Where does the 3 rd term come from? multiply the last terms of each bracket Where does the middle term come from?

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],[object Object],[object Object],(x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x = x 2 + 7x + 12 + x ×4 + 3×4 Where does the 3 rd term come from? multiply the last terms of each bracket Where does the middle term come from? add the last two terms of each bracket

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],x 2 + 5x + 6

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 5x + 6 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 5x + 6 Your task is to find two numbers so that when you multiply them you get the last term and when you add them you get the middle term + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 5x + 6 Your task is to find two numbers so that when you multiply them you get the last term and when you add them you get the middle term + + Your answer will always look like this For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) and added together gives 5 (circle the two numbers) write these two numbers in the brackets factors of 6 1 6 2 3

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 5x + 6 Your task is to find two numbers so that when you multiply them you get the last term and when you add them you get the middle term + 2 + 3 Your answer will always look like this For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) and added together gives 5 (circle the two numbers) write these two numbers in the brackets factors of 6 1 6 2 3

Factorising Quadratics Factorising Quadratic Expressioins ,[object Object],= (x )(x ) x 2 + 9x + 8 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressioins ,[object Object],= (x )(x ) x 2 + 9x + 8 Your task is to find two numbers so that their product is the last term and their sum is the middle term + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressioins ,[object Object],= (x )(x ) x 2 + 9x + 8 Your task is to find two numbers so that their product is the last term and their sum is the middle term + + Your answer will always look like this For this example you must find two numbers that multiplied together give 8 (write down the factors of 8) and added together gives 9 (circle the two numbers) write these two numbers in the brackets factors of 8 1 8 2 4

Factorising Quadratics Factorising Quadratic Expressioins ,[object Object],= (x )(x ) x 2 + 9x + 8 Your task is to find two numbers so that their product is the last term and their sum is the middle term + 1 + 8 Your answer will always look like this For this example you must find two numbers that multiplied together give 8 (write down the factors of 8) and added together gives 9 (circle the two numbers) write these two numbers in the brackets factors of 8 1 8 2 4

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 9x + 18 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 9x + 18 Your task is to find two numbers so that their product is the last term and their sum is the middle term + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 9x + 18 Your task is to find two numbers so that their product is the last term and their sum is the middle term + + Your answer will always look like this For this example you must find two numbers that multiplied together give 18 (write down the factors of 18) and added together gives 9 (circle the two numbers) write these two numbers in the brackets factors of 18 1 18 2 9 3 6

Factorising Quadratics Factorising Quadratic Expressions ,[object Object],= (x )(x ) x 2 + 9x + 18 Your task is to find two numbers so that their product is the last term and their sum is the middle term + 3 + 6 Your answer will always look like this For this example you must find two numbers that multiplied together give 18 (write down the factors of 18) and added together gives 9 (circle the two numbers) write these two numbers in the brackets factors of 18 1 18 2 9 3 6

Factorising Quadratics Predicting the signs ,[object Object],[object Object],Both + x 2 + 5x + 6 If the 2 nd sign is + the both signs of the brackets will be the SAME The 1 st sign tells you that both signs will be +. = (x + 2)(x + 3)

Factorising Quadratics Predicting the signs ,[object Object],[object Object],Both – x 2 – 5x + 6 If the 2 nd sign is + the both signs of the brackets will be the SAME The 1 st sign tells you that both signs will be – . = (x – 2)(x – 3)

Factorising Quadratics Predicting the signs ,[object Object],[object Object],Larger number is – x 2 – x – 6 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be – . = (x + 2)(x – 3)

Factorising Quadratics Predicting the signs ,[object Object],[object Object],Larger number is + x 2 + x – 6 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + . = (x – 2)(x + 3)

Factorising Quadratics Predicting the signs ,[object Object],[object Object],x 2 + x + 6 Both will be + (x + )(x + ) x 2 – x + 6 Both will be – (x – )(x – ) x 2 – x – 6 Larger number will be – Smaller number will be + (x + )(x – ) x 2 + x – 6 Larger number will be + Smaller number will be – (x – )(x + )

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 3x – 10 + – If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be – .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 3x – 10 + 2 – 5 For this example you must find two numbers that multiplied together give –10 (write down the factors of 10) and added together gives –3 : the larger number will be negative, the smaller will be positive (circle the two numbers) write these two numbers in the brackets factors of 10 1 –10 2 –5 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be – . = (x )(x ) + –

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 3x – 10 + 2 – 5 For this example you must find two numbers that multiplied together give –10 (write down the factors of 10) and added together gives –3 : the larger number will be negative, the smaller will be positive (circle the two numbers) write these two numbers in the brackets factors of 10 1 –10 2 –5 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be – . = (x )(x ) + 2 – 5

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 7x + 6 – – If the 2 nd sign is + the signs will be the SAME The 1 st sign tells you that both will be – .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 7x + 6 – – For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) and added together gives –7 : both numbers are negative (circle the two numbers) write these two numbers in the brackets factors of 6 – 1 –6 – 2 –3 If the 2 nd sign is + the signs will be the SAME The 1 st sign tells you that both will be – .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 – 7x + 6 – 1 – 6 For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) and added together gives –7 : both numbers are negative (circle the two numbers) write these two numbers in the brackets factors of 6 – 1 –6 – 2 –3 If the 2 nd sign is + the signs will be the SAME The 1 st sign tells you that both will be – .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + x – 12 – + If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + x – 12 – + For this example you must find two numbers that multiplied together give –12 (write down the factors of 12) and added together gives –1 : the larger number will be positive, the smaller will be negative (circle the two numbers) write these two numbers in the brackets factors of 12 – 1 12 – 2 6 – 3 4 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + x – 12 – 3 + 4 For this example you must find two numbers that multiplied together give –12 (write down the factors of 12) and added together gives +1 : the larger number will be positive, the smaller will be negative (circle the two numbers) write these two numbers in the brackets factors of 12 – 1 12 – 2 6 – 3 4 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + 2x – 8 – + If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + 2x – 8 – + For this example you must find two numbers that multiplied together give –8 (write down the factors of 8) and added together gives +2 : the larger number will be positive, the smaller will be negative (circle the two numbers) write these two numbers in the brackets factors of –8 – 1 8 – 2 4 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Predicting the signs ,[object Object],= (x )(x ) x 2 + 2x – 8 – 2 + 4 For this example you must find two numbers that multiplied together give –8 (write down the factors of 8) and added together gives +2 : the larger number will be positive, the smaller will be negative (circle the two numbers) write these two numbers in the brackets factors of –8 – 1 8 – 2 4 If the 2 nd sign is – the signs will be OPPOSITE The 1 st sign tells you that the larger factor will be + .

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 12x 2 + x – 6 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b E.g.1

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 12x 2 + x – 6 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b ac = 12 × –6 = –72 ac = –72 b = 1 E.g.1

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 12x 2 + x – 6 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of -72 – 1 72 – 2 36 – 3 14 – 4 18 – 6 12 – 8 9 ac = 12 × –6 = –72 ac = –72 b = 1 – 8 × 9 = – 72 – 8 + 9 = 1 p = –8 q = 9 E.g.1

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 12x 2 + x – 6 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of -72 – 1 72 – 2 36 – 3 14 – 4 18 – 6 12 – 8 9 ac = 12 × –6 = –72 ac = –72 b = 1 – 8 × 9 = – 72 – 8 + 9 = 1 p = –8 q = 9 = 12x 2 – 8x + 9x – 6 Now factorise ax 2 + px + qx +c in two stages 12x 2 + x – 6 E.g.1

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 12x 2 + x – 6 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of -72 – 1 72 – 2 36 – 3 14 – 4 18 – 6 12 – 8 9 ac = 12 × –6 = –72 ac = –72 b = 1 – 8 × 9 = – 72 – 8 + 9 = 1 p = –8 q = 9 = 12x 2 – 8x + 9x – 6 Now factorise ax 2 + px + qx +c in two stages = 4x(3x – 2) + 3(3x – 2) = (3x – 2)(4x + 3) 12x 2 + x – 6 E.g.1

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 3x 2 + 7 x + 2 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b E.g.2

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 3x 2 + 7 x + 2 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b ac = 3 × 2 = 6 ac = 6 b = 7 E.g.2

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 3x 2 + 7 x + 2 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 6 1 6 2 3 ac = 3 × 2 = 6 ac = 6 b = 7 1 × 6 = 6 1 + 6 = 7 p = 1 q = 6 E.g.2

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 3x 2 + 7 x + 2 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 6 1 6 2 3 ac = 3 × 2 = 6 ac = 6 b = 7 1 × 6 = 6 1 + 6 = 7 p = 1 q = 6 = 3x 2 + 1x + 6x + 2 Now factorise ax 2 + px + qx +c in two stages 3x 2 + 7 x + 2 E.g.2

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 3x 2 + 7 x + 2 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 6 1 6 2 3 ac = 3 × 2 = 6 ac = 6 b = 7 1 × 6 = 6 1 + 6 = 7 p = 1 q = 6 = 3x 2 + 1x + 6x + 2 Now factorise ax 2 + px + qx +c in two stages = x(3x + 1) + 2(3x + 1) = (3x + 1)(x + 2) 3x 2 + 7 x + 2 E.g.2

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 10x 2 – 13 x – 3 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b E.g.3

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 10x 2 – 13 x – 3 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b ac = 10 × –3 = –30 ac = –30 b = –13 E.g.3

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 10x 2 – 13 x – 3 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 30 1 –30 2 –15 3 –10 5 –6 ac = 10 × –3 = –30 ac = –30 b = –13 2 × –15 = –30 2 + –15 = –13 p = 2 q = –15 E.g.3

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 10x 2 – 13 x – 3 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 30 1 –30 2 –15 3 –10 5 –6 ac = 10 × –3 = –30 ac = –30 b = –13 2 × –15 = –30 2 + –15 = –13 p = 2 q = –15 = 10x 2 + 2x – 15x – 3 Now factorise ax 2 + px + qx +c in two stages 10x 2 – 13 x – 3 E.g.3

Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x 2 is not 1 10x 2 – 13 x – 3 For the equation ax 2 + bx +c Your task is to find two numbers (call them p and q ) so that when you multiply them you get the ac and when you add them you get the b factors of 30 1 –30 2 –15 3 –10 5 –6 ac = 10 × –3 = –30 ac = –30 b = –13 2 × –15 = –30 2 + –15 = –13 p = 2 q = –15 = 10x 2 + 2x – 15x – 3 Now factorise ax 2 + px + qx +c in two stages = 2x(5x + 1) – 3(5x + 1) = (5x + 1)(2x – 3 ) 10x 2 – 13 x – 3 E.g.3

Factorising Quadratics

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Factorising Quadratics