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? Remember expanding two bracket problems

Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x × x

Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x × x + x ×4

Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x × x + 3 ×x + x ×4

Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x × x + 3 ×x + x ×4 + 3×4

Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x × x = x 2 + 4x + 3x + 12 + 3 ×x + x ×4 + 3×4

Factorising Quadratics What are they? Remember expanding two bracket problems (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? Remember expanding two bracket problems (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 Factorising is reversing the process of removing brackets. To factorise a quadratic you need to put back the two brackets. Let’s take a closer look at how the quadratic was formed. (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 Factorising is reversing the process of removing brackets. To factorise a quadratic you need to put back the two brackets. Let’s take a closer look at how the quadratic was formed. (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 Factorising is reversing the process of removing brackets. To factorise a quadratic you need to put back the two brackets. Let’s take a closer look at how the quadratic was formed. (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 Factorising is reversing the process of removing brackets. To factorise a quadratic you need to put back the two brackets. Let’s take a closer look at how the quadratic was formed. (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 Factorising is reversing the process of removing brackets. To factorise a quadratic you need to put back the two brackets. Let’s take a closer look at how the quadratic was formed. (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 Now let’s start with a quadratic equation and try to find the two brackets x 2 + 5x + 6

Factorising Quadratics Factorising Quadratic Expressions Now let’s start with a quadratic equation and try to find the two brackets = (x )(x ) x 2 + 5x + 6 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions Now let’s start with a quadratic equation and try to find the two brackets = (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 Now let’s start with a quadratic equation and try to find the two brackets = (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 Now let’s start with a quadratic equation and try to find the two brackets = (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 Example 2: Factorise the following quadratic equation = (x )(x ) x 2 + 9x + 8 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressioins Example 2: Factorise the following quadratic equation = (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 Example 2: Factorise the following quadratic equation = (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 Example 2: Factorise the following quadratic equation = (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 Example 3: Factorise the following quadratic equation = (x )(x ) x 2 + 9x + 18 + + Your answer will always look like this

Factorising Quadratics Factorising Quadratic Expressions Example 3: Factorise the following quadratic equation = (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 Example 3: Factorise the following quadratic equation = (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 Example 3: Factorise the following quadratic equation = (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 What happens when there are negative numbers in the equation? Here are the various options: First look at the 2 nd sign, then the 1 st sign. 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 What happens when there are negative numbers in the equation? Here are the various options: First look at the 2 nd sign, then the 1 st sign. 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 What happens when there are negative numbers in the equation? Here are the various options: First look at the 2 nd sign, then the 1 st sign. 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 What happends when there are negative numbers in the equation? Here are the various options: First look at the 2 nd sign, then the 1 st sign. 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 What happends when there are negative numbers in the equation? Here are the various options: First look at the 2 nd sign, then the 1 st sign. 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 Example 4: Factorise the following quadratic equation = (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 Example 4: Factorise the following quadratic equation = (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 Example 4: Factorise the following quadratic equation = (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 Example 4: Factorise the following quadratic equation = (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 Example 4: Factorise the following quadratic equation = (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 Example 4: Factorise the following quadratic equation = (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 Example 6: Factorise the following quadratic equation = (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 Example 6: Factorise the following quadratic equation = (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 Example 6: Factorise the following quadratic equation = (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 Example 7: Factorise the following quadratic equation = (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 Example 7: Factorise the following quadratic equation = (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 Example 7: Factorise the following quadratic equation = (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 Method: Using the quadratic can be written as ax 2 + bx +c 1. Look for two numbers that: multiply to ac and add to b Call these numbers p and q 2. Write ax 2 + bx +c as ax 2 + px + qx +c 3. Now factorise ax 2 + px + qx +c in two stages

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