1. www.mathsrevision.com S4 Credit Algebraic Operations Introduction to Quadratic Equation Summary of Factorising Methods Factorising Trinomials (Quadratics) Real-life Problems on Quadratics 18-Jun-16Created by Mr. Lafferty@mathsrevision.com Finding roots by factorising and formula Exam Type Questions

2. 18-Jun-16 Starter Questions www.mathsrevision.com Q1.Remove the brackets (a)a (4y – 3x) =(b)(x + 5)(x - 5) = Q2.For the line y = -x + 5, find the gradient and where it cuts the y axis. S4 Credit Created by Mr. Lafferty@mathsrevision.com Q3.Find the highest common factor for p 2 q and pq 2.

3. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.To be able to identify the three methods of factorising. 1.To review the three basic methods for factorising. 2.Apply knowledge to problems. www.mathsrevision.com Factorising Methods S4 Credit

4. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Summary of Factorising When we are asked to factorise there is priority we must do it in. 1.Take any common factors out and put them outside the brackets. 2.Check for the difference of two squares. 3.Factorise any quadratic expression left. S4 Credit

5. 18-Jun-16Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Common Factor Factorise the following : (a) 4xy – 2x (b)y 2 - y 2x(y – 1) y(y – 1) S4 Credit

6. 18-Jun-16Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com When we have the special case that an expression is made up of the difference of two squares then it is simple to factorise The format for the difference of two squares a 2 – b 2 First square term Second square term Difference Difference of Two Squares S4 Credit

7. 18-Jun-16Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com a 2 – b 2 First square term Second square term Difference This factorises to ( a + b )( a – b ) Two brackets the same except for + and a - Check by multiplying out the bracket to get back to where you started Difference of Two Squares S4 Credit

8. 18-Jun-16Created by Mr. Lafferty www.mathsrevision.com Keypoints Formata 2 – b 2 Always the difference sign - ( a + b )( a – b ) Difference of Two Squares S4 Credit

9. 18-Jun-16Created by Mr. Lafferty www.mathsrevision.com Factorise using the difference of two squares (a) w 2 – z 2 (b) 9a 2 – b 2 (c)16y 2 – 100k 2 ( w + z )( w – z ) ( 3a + b )( 3a – b ) ( 4y + 10k )( 4y – 10k ) Difference of Two Squares S4 Credit

10. 18-Jun-16Created by Mr. Lafferty www.mathsrevision.com Factorise these trickier expressions. (a)6x 2 – 24 (b) 3w 2 – 3 (c) 8 – 2b 2 (d) 27w 2 – 12 6(x + 2 )( x – 2 ) 3( w + 1 )( w – 1 ) 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) Difference of Two Squares S4 Credit

11. www.mathsrevision.com S4 Credit + 1 + 2+ 2 18-Jun-16Created by Mr. Lafferty@mathsrevision.com x 2 + 3x + 2 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 Find two numbers that multiply to give last number (+2) and Diagonals sum to give middle value +3x. ( ) x x (+2) x ( +1) = +2 (+2x) +( +1x) = +3x

12. www.mathsrevision.com S4 Credit + 5 + 1 + 5 18-Jun-16Created by Mr. Lafferty@mathsrevision.com x 2 + 6x + 5 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 ( ) x x Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6x. (+5) x ( +1) = +5 (+5x) +( +1x) = +6x

13. www.mathsrevision.com S4 Credit - 2 - 2- 2 - 2 18-Jun-16Created by Mr. Lafferty@mathsrevision.com x 2 - 4x + 4 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x ( ) x x Both numbers must be - Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4x. (-2) x ( -2) = +4 (-2x) +( -2x) = -4x

14. www.mathsrevision.com S4 Credit - 3- 3 + 1 18-Jun-16Created by Mr. Lafferty@mathsrevision.com x 2 - 2x - 3 Strategy for factorising quadratics Factorising Using St. Andrew’s Cross method x x+ 1 ( ) x x One number must be + and one - Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2x (-3) x ( +1) = -3 (-3x) +( x) = -2x

15. 18-Jun-16Created by Mr. Lafferty www.mathsrevision.com Factorise using SAC method (a)m 2 + 2m +1 (b) y 2 + 6m + 5 (c) b 2 – b -2 (d)a 2 – 5a + 6 (m + 1 )( m + 1 ) ( y + 5 )( y + 1 ) ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) Factorising Using St. Andrew’s Cross method S4 Credit

16. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 1.1 Ch8 (page156) www.mathsrevision.com Factorising Methods S4 Credit

17. Starter Questions 18-Jun-16Created by Mr. Lafferty@mathsrevision.com www.mathsrevision.com Q1.True or false y ( y + 6 ) -7y = y 2 -7y + 6 Q2.Fill in the ? 49 – 4x 2 = ( ? + ?x)(? – 2?) Q3.Write in scientific notation 0.0341 S4 Credit

18. www.mathsrevision.com S4 Credit Quadratic Equations A quadratic function has the form f(x) = a x 2 + b x + c The graph of a quadratic function has the basic shape The x-coordinates where the graph cuts the x – axis are called the Roots of the function. y x i.e. a x 2 + b x + c = 0 a, b and c are constants and a ≠ 0 y x This is called a quadratic equation

19. www.mathsrevision.com S4 Credit Quadratic Equations This is the graph of a golf shot. The height h m of the ball after t seconds is given by : h = 15t – 5t 2 h t The graph of a quadratic function is called a parabola (a) For what values t does h = 0 (b) What are the solutions for 15t – 5t 2 = 0 t = 0t = 3 t = 0t = 3and

20. www.mathsrevision.com S4 Credit Quadratic Equations This is the graph of a parabola h = 10t – 2t 2 h t (a) From the graph, what are the roots of the quadratic eqn. (b) What is the value of h for t = 1 and t = 4 10t – 2t 2 = 0 t = 0 t = 5 h = 10t – 2t 2 (c) What are the solutions of the quadratic equation (d) What is the solution of the quadratic equation 10t – 2t 2 = 12.5 Both 8 t = 0t = 5and 2.5

21. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex2.1 Q2 & Q4 Ch8 (page 158) www.mathsrevision.com Quadratic Equation S4 Credit

22. 18-Jun-16 Starter Questions www.mathsrevision.com Q1.Multiple out the brackets and simplify. (a)( 2x – 5 )( x + 5 ) Created by Mr. Lafferty@mathsrevision.com Q3.Find the gradient and where line cut y-axis. x = y + 1 Q2.Find the volume of a cylinder with high 6m and diameter 9cm S4 Credit

23. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Learning Intention Success Criteria 1.Be able find factors using the three methods to solve quadratic equations. 1.To explain how factors help to solve quadratic equations. www.mathsrevision.com Factors and Solving Quadratic Equations S4 Credit

24. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com The main reason we learn the process of factorising is that it helps to solve (find roots) for quadratic equations. 1.Take any common factors out and put them outside the brackets. 2.Check for the difference of two squares. 3.Factorise any quadratic expression left. S4 Credit Factors and Solving Quadratic Equations Reminder of Methods

25. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following x 2 – 4x = 0 x(x – 4) = 0 x = 0and x - 4 = 0 x = 4 16t – 6t 2 = 0 4t(8 – 3t) = 0 4t = 0and8 – 3t = 0 t = 8/3t = 0and Common Factor Common Factor

26. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following x 2 – 9 = 0 (x – 3)(x + 3) = 0 x = 3and x = -3 100s 2 – 25 = 0 (10s – 5)(10s + 5) = 0 10s – 5 = 0and10s + 5 = 0 s = - 0.5s = 0.5and Difference 2 squares Difference 2 squares

27. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 3.1 Ch8 (page 159) www.mathsrevision.com S4 Credit Factors and Solving Quadratic Equations

28. www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples 2x 2 – 8 = 0 2(x 2 – 4) = 0 x = 2andx = - 2 80 – 125e 2 = 0 5(16 – 25e 2 ) = 0 4 – 5e = 0and4 + 5t = 0 t = - 4/5t = 4/5and Common Factor Common Factor Difference 2 squares 2(x – 2)(x + 2) = 0 (x – 2)(x + 2) = 0 Difference 2 squares 5(4 – 5e)(4 + 5e) = 0 (4 – 5e)(4 + 5e) = 0 (x – 2) = 0and(x + 2) = 0

29. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 3.2 Ch8 (page 160) www.mathsrevision.com S4 Credit Factors and Solving Quadratic Equations

30. www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following x 2 + 5x + 4 = 0 (x + 4)(x + 1) = 0 x = - 4andx = - 1 SAC Method x x 4 1 x + 4 = 0x + 1 = 0and 1 + x - 6x 2 = 0 (1 + 3x)(1 – 2x) = 0 x = - 1/3andx = 0.5 SAC Method 1 1 +3x -2x 1 + 3x = 0and1 - 2x = 0

31. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 4.1 Ch8 (page 161) www.mathsrevision.com S4 Credit Factors and Solving Quadratic Equations

32. www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following (x + 4) 2 =36 (x + 10)(x - 2) = 0 x = - 10andx = - 2 Multiply out and rearrange x x 10 - 2 x + 10 = 0x - 2 = 0and 5x(2x + 1) - 10 = x(7x + 6) (3x + 5)(x – 2) = 0 x = - 5/3and SAC Method 3x x +5 -2 3x + 5 = 0andx - 2 = 0 x 2 + 8x - 20 = 0 SAC Method Multiply out and rearrange 3x 2 - x - 10 = 0 x = 2

33. www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following (x - 4)(x + 1) = 0 x = 4andx = - 1 Multiply through by 2(x - 1)(x + 2) to remove denominators x x - 4 1 x - 4 = 0x + 1 = 0and 2(x + 2) + 2(x – 1) = (x – 1)(x + 2) SAC Method 2x + 4 + 2x – 2 = x 2 + x - 2 x 2 - 3x – 4 = 0

34. www.mathsrevision.com Solving Quadratic Equations S4 Credit Examples Solve ( find the roots ) for the following (x + 3)(x - 2) = 0 x = - 3andx = 2 Multiply through by x(x + 1) to remove denominators x x 3 - 2 x + 3 = 0x - 2 = 0and 6(x + 1) - 6x = x(x + 1) SAC Method 6x + 6 – 6x = x 2 + x x 2 + x – 6 = 0

35. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 4.2 Ch8 (page 162) www.mathsrevision.com S4 Credit Factors and Solving Quadratic Equations

36. created by Mr. Lafferty Starter Questions www.mathsrevision.com S4 Credit

37. created by Mr. Lafferty Learning Intention Success Criteria 1.To be able to using quadratic theory in real-life problem. 1.To show how quadratic theory is used in real- life. www.mathsrevision.com S4 Credit Real-life Quadratics

38. www.mathsrevision.com S4 Credit Real-life Problems A rectangle garden is twice as long as it is wide. The area is 200m 2. Find the dimensions of the rectangle garden. Let width be x Length is 2 x Area = length x breadth 200 = 2 x x x 200 = 2 x 2 x 2 = 100 x = 10 and x = -10 x must be positive ( We cannot get a negative length !!! ) Width is equal to 10mLength is equal to 20m

39. www.mathsrevision.com S4 Credit Real-life Problems The height in metres of a rocket fired vertically upwards is give by the formula : (a) When will the rocket be at a height of 160 metres. h = 176t – 16t 2 160 = 176t – 16t 2 16t 2 - 176t + 160 = 0 t 2 - 11t + 10 = 0 (t – 10)(t – 1) = 0 t = 10andt = 1 (b) Is it possible for the rocket to h = 188 metres. Since 188 = 176t -16t 2 has no solution not possible.

40. 18-Jun-16 Created by Mr. Lafferty@www.mathsrevision.com Now try MIA Ex 5.1 & 5.2 Ch8 (page 164) www.mathsrevision.com S4 Credit Real-life Quadratics

41. created by Mr. Lafferty Starter Questions www.mathsrevision.com S4 Credit

42. created by Mr. Lafferty Learning Intention Success Criteria 1.To be able to solve quadratic equations using quadratic formula. 1.To explain how to find the roots (solve) quadratic equations by use quadratic formula. www.mathsrevision.com S4 Credit Roots Formula

43. created by Mr. Lafferty www.mathsrevision.com Every quadratic equation can be rearranged into the standard form Roots Formula ax 2 + bx + c = 0 S4 Credit a, b and c are constants Examples : find the constants a, b and c for the following 3x 2 + x + 4 = 0a = 3b = 1c = 4 x 2 - x - 6 = 0a = 1b = -1c = -6 x(x - 2) = 0x 2 – 2x = 0a = 1b = -2c = 0

44. created by Mr. Lafferty Now try MIA Ex6.1 First Column (page 166) www.mathsrevision.com S4 Credit Roots Formula

45. created by Mr. Lafferty www.mathsrevision.com Every quadratic equation can be rearranged into the standard form Roots Formula ax 2 + bx + c = 0 S4 Credit a, b and c are constants In this form we can using the quadratic root formula to find the roots.

46. created by Mr. Lafferty www.mathsrevision.com Example : Solve x 2 + 3x - 3 ax 2 + bx + c 13-3 S4 Credit Roots Formula

47. created by Mr. Lafferty www.mathsrevision.com and S4 Credit and Roots Formula

48. www.mathsrevision.com S4 Credit created by Mr. Lafferty Use the quadratic formula to solve the following : 2x 2 + 4x + 1 = 0 5x 2 - 9x + 3 = 0 3x 2 - 3x – 5 = 0 x 2 + 3x – 2 = 0 x = 1.9, -0.9 x = -1.7, -0.3 x = -3.6, 0.6 x = 1.4, 0.4 Roots Formula

49. created by Mr. Lafferty Now try MIA Ex7.1 & 7.2 (page 168) www.mathsrevision.com S4 Credit Roots Formula

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