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Factorisation. Single Brackets.

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monicai
13 diapositives · 204 Views · Jan 04, 2020

Learn how to factorise quadratics with simple examples and detailed explanations. Discover how to factorise by single brackets and put double brackets back in. Practice examples for better understanding.

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  1. Factorisation. Single Brackets. Multiply out the bracket below: 2x ( 4x – 6 ) Example 2 Factorise: 40x2 – 5x = 8x2 - 12x = 5 ( 8x2 - x ) Factorisation is the reversal of the above process. That is to say we put the brackets back in. = 5 x ( 8 x - 1 ) Example 1 Factorise: 4x2 – 12 x Hint:Numbers First. = 4 ( x2 – 3x) Hint:Now Letters = 4x ( x – 3 )

  2. What Goes In The Box ? Factorise fully : 12 x2 – 6 x Now factorise the following: (1) 14 x 2 + 7 x =7x( x + 1) 6 ( 2x2 - x ) (2) 4x – 12 x 2 = 4x ( 1 – 3x) 6x ( 2x - 1 ) (3) 6ab – 2ad =2a( 3b – d) (4) 12 a2 b – 6 a b2 = 6ab ( a – b)

  3. A Difference Of Two Squares. Consider what happens when you multiply out : ( x + y ) ( x – y) Now you try the example below: Example. Multiply out: ( 5 x + 7 y )( 5 x – 7 y ) = x ( x – y ) + y ( x – y ) - y 2 =x 2 - xy + xy Answer: = x2 - y2 = 25 x 2 - 49 y 2 This is a difference of two squares.

  4. What Goes In The Box ? Mutiply out: (1) ( 3 x + 6 y ) ( 3 x – 6 y) (4) ( x – 11 y ) ( x + 11 y) = 9 x 2 – 36 y 2 = x 2 – 121 y 2 (2) ( 2 x – 4 y ) ( 2 x + 4 y) (5) ( 7 x + 2 y ) ( 7 x – 2 y) = 4 x 2 – 16 y 2 = 49 x 2 – 4 y 2 (6) ( 5 x – 9 y ) ( 5 x + 9 y) (3) ( 8 x + 9 y ) ( 8 x – 9 y) = 64 x 2 – 81 y 2 – 81 y 2 = 25 x 2 (7) ( 3 x + 9 y ) ( 3 x – 9 y) (3) ( 5 x – 7 y ) ( 5 x + 7 y) = 9 x 2 – 81 y 2 = 25 x 2 – 49 y 2

  5. Factorising A Difference Of Two Squares. By considering the brackets required to produce the following factorise the following examples directly: Examples (5) 4x 2 - 36 (1) x 2 - 9 = ( 2x - 6 ) ( 2x + 6 ) = ( x - 3 ) ( x + 3 ) (2) x 2 - 16 (6) 9x 2 - 16y 2 = ( x - 4 ) ( x + 4 ) = ( 3x - 4y ) ( 3x + 4y ) (3) x 2 - 25 (7) 100g 2 - 49k 2 = ( 10 g – 7k ) ( 10g + 7k ) = ( x - 5 ) ( x + 5 ) (8) 144d 2 - 36w 2 (4) x 2 - y 2 = ( 12d - 6 w) ( 12d + 6w ) = ( x - y ) ( x + y )

  6. What Goes In The Box ? Multiply out the brackets below: (3x – 4 ) ( 2x + 7) 3x (2x + 7) -4 (2x + 7) 6x 2 -8x -28 +21x You are now about to discover how to put the double brackets back in. 6x 2 -28 +13x

  7. Follow the steps below to put a double bracket back into a quadratic equation. Factorising A Quadratic. Process. Step 1: Consider the factors of the coefficient in front of the x and the constant. Factorise the quadratic: x2 – 2x - 15 Factors 5x 1 15 Step 2 : Create the x coefficient from two pairs of factors. = (x 5) ( x 3) 1 1 1 15 3x 3 5 Step 3 Place the four numbers in the pair of brackets looking at outer and inner pairs to determine the signs. 3x – 5x = - 2x x coefficient = 2 (1 x 5) – (1 x 3 ) = 2 = (x - 5) ( x +3)

  8. More Quadratic Factorisation Examples. Example 1. Factorise the quadratic: x2 + 3x - 10 Factors 1 10 1 1 1 10 5x = (x 5) ( x 2) 2 5 2x x coefficient = 3 (1 x 5) - (1 x 2 ) = 3 = (x + 5) ( x - 2 ) Signs in brackets. 5x – 2x = 3x

  9. Quadratic Factorisation Example 2 Factorise the quadratic: x2 – 8x + 12 Factors 1 12 1 1 1 12 6x = (x 6) ( x 2) 2 6 2x 3 4 x coefficient = 8 = (x - 6) ( x -2 ) (1 x 6) + (1 x 2 ) = 8 Signs in brackets. - 6x – 2x = - 8x

  10. Quadratic Factorisation Example 3. Factorise the quadratic: 6 x2 + 11x – 10 Factors 6 10 1 6 1 10 4x = (3x 2) ( 2x 5) 2 3 2 5 15x x coefficient = 11 = ( 3x - 2) ( 2 x + 5) (3 x 5) – (2 x 2 ) = 11 Numbers together. Signs in brackets. Numbers apart. 15 x – 4x = 11x

  11. Quadratic Factorisation Example 4 Factorise the quadratic: 10 x2 + 27x – 28 Factors 10 28 1 10 1 28 8x 2 5 2 14 = (5x 4) ( 2x 7) 4 7 35x x coefficient = 27 = ( 5x - 4) ( 2 x + 7) (5 x 7) – (2 x 4 ) = 27 Signs in brackets. 35 x – 8x = 27x

  12. What Goes In The Box ? Factorise the quadratic: 6 x2 – x – 2 Factors 6 2 1 6 1 2 2 3 = (3x 2) ( 2x 1) x coefficient -1 = ( 3x - 2) ( 2 x + 1) (2 x 2) – (1 x 3 ) = 1 Signs in brackets. 3 x – 4x = -x

  13. What Goes In The Box 2 Factorise the quadratic: 15 x2 – 19x + 6 Factors 15 6 1 15 1 6 3 5 2 3 = (3x 2) ( 5x 3) x coefficient -19 = ( 3x - 2) ( 5 x - 3) (3 x 3) + (5 x 2 ) = 19 Signs in brackets. - 9 x – 10x = - 19x