Factorising Quadratics

Slide 1: Factorising Quadratics Index 1. What are quadratics? 2. Factorising quadratics (coefficient of x2 is 1) 3. Predicting the signs of the final answer. 4. Factorising Quadratics (coefficient of x2 is not 1)

Slide 2: Factorising Quadratics What are they? Remember expanding two bracket problems (x + 3)(x + 4) = x×x + x×4 + 3×x + 3×4 = x2 + 4x + 3x + 12 add the like terms = x2 + 7x + 12 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: x2 + x + 2 x2 – 2x + 6 4x2 – 100 2x2 – 14x + 20 Back to INDEX page or press spacebar to continue CM ppt

Slide 3: 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) Where does the 3rd term come from? multiply the last terms of each bracket = x×x + x×4 + 3×x + 3×4 Where does the middle term come from? = x2 + 4x + 3x + 12 add the last two terms of each bracket = x2 + 7x + 12 CM ppt

Slide 4: Factorising Quadratics Factorising Quadratic Expressions Now let’s start with a quadratic equation and try to find the two brackets Your answer will always look like this x2 + 5x + 6 Your task is to find two numbers so that when you multiply them you get the last term = (x + 2 )(x + 3 ) and when you add them you get the middle term factors of 6 For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) 1 6 and 2 3 added together gives 5 (circle the two numbers) write these two numbers in the brackets CM ppt

Slide 5: Factorising Quadratics Factorising Quadratic Expressioins Example 2: Factorise the following quadratic equation Your answer will always look like this x2 + 9x + 8 Your task is to find two numbers so that their product is the last term = (x + 1 )(x + 8 ) and their sum is the middle term factors of 8 For this example you must find two numbers that multiplied together give 8 (write down the factors of 8) 1 8 and 2 4 added together gives 9 (circle the two numbers) write these two numbers in the brackets CM ppt

Slide 6: Factorising Quadratics Factorising Quadratic Expressions Example 3: Factorise the following quadratic equation Your answer will always look like this x2 + 9x + 18 Your task is to find two numbers so that their product is the last term = (x + 3 )(x + 6 ) and their sum is the middle term factors of 18 For this example you must find two numbers that multiplied together give 18 (write down the factors of 18) 1 18 and 2 9 added together gives 9 (circle the two numbers) 3 6 write these two numbers in the brackets Back to INDEX page or press spacebar to continue CM ppt

Slide 7: Factorising Quadratics Predicting the signs What happens when there are negative numbers in the equation? Here are the various options: First look at the 2nd sign, then the 1st sign. If the 2nd sign is + the both signs of the brackets will be the SAME x2 + 5x + 6 = (x + 2)(x + 3) The 1st sign tells you Both + that both signs will be +. CM ppt

Slide 8: Factorising Quadratics Predicting the signs What happens when there are negative numbers in the equation? Here are the various options: First look at the 2nd sign, then the 1st sign. If the 2nd sign is + the both signs of the brackets will be the SAME x2 – 5x + 6 = (x – 2)(x – 3) The 1st sign tells you Both – that both signs will be – . CM ppt

Slide 9: Factorising Quadratics Predicting the signs What happens when there are negative numbers in the equation? Here are the various options: First look at the 2nd sign, then the 1st sign. If the 2nd sign is – the signs will be OPPOSITE x2 – x – 6 = (x + 2)(x – 3) Larger The 1st sign tells you number that the larger factor will is – be – . CM ppt

Slide 10: Factorising Quadratics Predicting the signs What happends when there are negative numbers in the equation? Here are the various options: First look at the 2nd sign, then the 1st sign. If the 2nd sign is – the signs will be OPPOSITE x2 + x – 6 = (x – 2)(x + 3) The 1st sign tells you Larger that the larger factor will number be + . is + CM ppt

Slide 11: Factorising Quadratics Predicting the signs What happends when there are negative numbers in the equation? Here are the various options: First look at the 2nd sign, then the 1st sign. x2 + x + 6 (x + )(x + ) Both will be + x2 – x + 6 (x – )(x – ) Both will be – Larger number will be – x2 – x – 6 (x + )(x – ) Smaller number will be + Larger number will be + x2 + x – 6 (x – )(x + ) Smaller number will be – CM ppt

Slide 12: Factorising Quadratics Predicting the signs Example 4: Factorise the following quadratic equation If the 2nd sign is – the signs will be OPPOSITE Your answer will always look like this x2 – 3x – 10 The 1st sign tells you = (x + 2)(x – 5 ) For this example you must find two numbers that that the larger factor will multiplied together give –10 (write down the factors of 10) be – . and added together gives –3 : the larger number will be negative, the smaller will be positive (circle the two numbers) factors of 10 write these two numbers in the brackets 1 –10 2 –5 CM ppt

Slide 13: Factorising Quadratics Predicting the signs Example 4: Factorise the following quadratic equation If the 2nd sign is + the signs will be the SAME Your answer will always look like this x2 – 7x + 6 The 1st sign tells you = that– 1)(x – 6 ) (x both will be –. For this example you must find two numbers that multiplied together give 6 (write down the factors of 6) and added together gives –5 : both numbers are negative factors of 6 (circle the two numbers) –1 –6 write these two numbers in the brackets –2 –3 CM ppt

Slide 14: Factorising Quadratics Predicting the signs Example 6: Factorise the following quadratic equation If the 2nd sign is – the signs will be OPPOSITE Your answer will always look like this x2 + x – 12 The 1st sign tells you = (x – 3)(x + 4 ) For this example you must find two numbers that that the larger factor will multiplied together give –12 (write down the factors of 12) be + . and added together gives –1 : the larger number will be positive, the smaller will be negative (circle the two numbers) factors of 12 write these two numbers in the brackets –1 12 –2 6 –3 4 CM ppt

Slide 15: Factorising Quadratics Predicting the signs Example 7: Factorise the following quadratic equation If the 2nd sign is – the signs will be OPPOSITE Your answer will always look like this x2 + 2x – 8 The 1st sign tells you = (x – 2)(x + 4 ) For this example you must find two numbers that that the larger factor will multiplied together give –8 (write down the factors of 8) be + . and added together gives 2 : the larger number will be factors of –8 positive, the smaller will be negative (circle the two –1 8 numbers) write these two numbers in the brackets –2 4 Back to INDEX page or press spacebar to continue CM ppt

Slide 16: Factorising Quadratics Factorising Quadratic Expressions where the coefficient of x2 is not 1 Method: Using the quadratic can be written as ax2 + bx +c 1. Look for two numbers that: • multiply to ac and • add to b Call these numbers p and q 2. Write ax2 + bx +c as ax2 + px + qx +c 3. Now factorise ax2 + px + qx +c in two stages CM ppt

Slide 17: Factorising Quadratics E.g.1 Factorising Quadratic Expressions where the coefficient of x2 is not 1 For the equation ax2 + bx +c ac = 12 × –6 = –72 Your task is to find two numbers (call 12x2 + x – 6 them p and q) so that when you multiply them you get the ac b=1 ac = –72 and when you add them you get the b factors of -72 ax2 + px + qx +c –8 × 9 = – 72 Now factorise –1 72 –2 36 in two stages –8+9=1 –3 14 12x2 + x – 6 –4 18 p = –8 –6 12 q=9 = 12x2 – 8x + 9x – 6 –8 9 = 4x(3x – 2) + 3(3x – 2) = (3x – 2)(4x + 3)

Slide 18: Factorising Quadratics E.g.2 Factorising Quadratic Expressions where the coefficient of x2 is not 1 For the equation ax2 + bx +c ac = 3 × 2 = 6 Your task is to find two numbers (call 3x2 + 7 x + 2 them p and q) so that when you multiply them you get the ac b=7 ac = 6 and when you add them you get the b factors of 6 ax2 + px + qx +c 1×6=6 Now factorise 1 6 2 3 in two stages 1+6=7 3x2 + 7 x + 2 p=1 q=6 = 3x2 + 1x + 6x + 2 = x(3x + 1) + 2(3x + 1) = (3x + 1)(x + 2)

Slide 19: Factorising Quadratics E.g.3 Factorising Quadratic Expressions where the coefficient of x2 is not 1 For the equation ax2 + bx +c ac = 10 × –3 = –30 Your task is to find two numbers (call 10x2 – 13 x – 3 them p and q) so that when you multiply them you get the ac b = –13 ac = –30 and when you add them you get the b factors of 30 ax2 + px + qx +c 2 × –15 = –30 Now factorise 1 –30 2 –15 in two stages 3 –10 2 + –15 = –13 10x2 – 13 x – 3 5 –6 p=2 q = –15 = 10x2 + 2x – 15x – 3 = 2x(5x + 1) – 3(5x + 1) Back to INDEX page = (5x + 1)(2x – 3 ) or press spacebar to end