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Algebraic Operations

Learn about the three basic methods of factorising algebraic expressions, including factorising by removing brackets, using the difference of two squares, and using St. Andrew's Cross method. Includes examples and practice problems.

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Algebraic Operations

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

  2. S4 Credit Q1. Remove the brackets (a) a (4y – 3x) = (b) (x + 5)(x - 5) = Starter Questions Q2. For the line y = -x + 5, find the gradient and where it cuts the y axis. www.mathsrevision.com Q3. Find the highest common factor for p2q and pq2. Created by Mr. Lafferty@mathsrevision.com

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

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

  5. Common Factor S4 Credit Factorise the following : 2x(y – 1) (a) 4xy – 2x (b) y2 - y www.mathsrevision.com y(y – 1) Created by Mr. Lafferty@mathsrevision.com

  6. Difference of Two Squares S4 Credit 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 www.mathsrevision.com a2 – b2 First square term Second square term Difference Created by Mr. Lafferty@www.mathsrevision.com

  7. Difference of Two Squares Check by multiplying out the bracket to get back to where you started S4 Credit a2 – b2 First square term Second square term Difference This factorises to www.mathsrevision.com ( a + b )( a – b ) Two brackets the same except for + and a - Created by Mr. Lafferty@www.mathsrevision.com

  8. Difference of Two Squares S4 Credit Keypoints Format a2 – b2 www.mathsrevision.com Always the difference sign - ( a + b )( a – b ) Created by Mr. Lafferty

  9. Difference of Two Squares S4 Credit Factorise using the difference of two squares ( w + z )( w – z ) • (a) w2 – z2 • (b) 9a2 – b2 • (c) 16y2 – 100k2 ( 3a + b )( 3a – b ) www.mathsrevision.com ( 4y + 10k )( 4y – 10k ) Created by Mr. Lafferty

  10. Difference of Two Squares S4 Credit Factorise these trickier expressions. 6(x + 2 )( x – 2 ) • (a) 6x2 – 24 • 3w2 – 3 • 8 – 2b2 • (d) 27w2 – 12 3( w + 1 )( w – 1 ) www.mathsrevision.com 2( 2 + b )( 2 – b ) 3(3 w + 2 )( 3w – 2 ) Created by Mr. Lafferty

  11. Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+2) and Diagonals sum to give middle value +3x. x2 + 3x + 2 x x + 2 + 2 (+2) x( +1) = +2 x x + 1 + 1 (+2x) +( +1x) = +3x ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  12. Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+5) and Diagonals sum to give middle value +6x. x2 + 6x + 5 x x + 5 + 5 (+5) x( +1) = +5 x x + 1 + 1 (+5x) +( +1x) = +6x ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  13. Both numbers must be - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (+4) and Diagonals sum to give middle value -4x. x2 - 4x + 4 x x - 2 - 2 (-2) x( -2) = +4 x x - 2 - 2 (-2x) +( -2x) = -4x ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  14. One number must be + and one - Factorising Using St. Andrew’s Cross method Strategy for factorising quadratics Find two numbers that multiply to give last number (-3) and Diagonals sum to give middle value -2x x2 - 2x - 3 x x - 3 - 3 (-3) x( +1) = -3 x x + 1 + 1 (-3x) +( x) = -2x ( ) ( ) Created by Mr. Lafferty@mathsrevision.com

  15. Factorising Using St. Andrew’s Cross method S4 Credit Factorise using SAC method (m + 1 )( m + 1 ) • (a) m2 + 2m +1 • y2 + 6m + 5 • b2 – b -2 • (d) a2 – 5a + 6 ( y + 5 )( y + 1 ) www.mathsrevision.com ( b - 2 )( b + 1 ) ( a - 3 )( a – 2 ) Created by Mr. Lafferty

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

  17. S4 Credit Q1. True or false y ( y + 6 ) -7y = y2 -7y + 6 Starter Questions Q2. Fill in the ? 49 – 4x2 = ( ? + ?x)(? – 2?) www.mathsrevision.com Q3. Write in scientific notation 0.0341 Created by Mr. Lafferty@mathsrevision.com

  18. This is called a quadratic equation Quadratic Equations S4 Credit A quadratic function has the form a , b and c are constants and a ≠ 0 f(x) = a x2 + b x + c The graph of a quadratic function has the basic shape www.mathsrevision.com y y The x-coordinates where the graph cuts the x – axis are called the Roots of the function. x x i.e. a x2 + b x + c = 0

  19. Quadratic Equations h This is the graph of a golf shot. The height h m of the ball after t seconds is given by : h = 15t – 5t2 The graph of a quadratic function is called a parabola (a) For what values t does h = 0 t (b) What are the solutions for t = 0 t = 3 15t – 5t2 = 0 t = 0 and t = 3

  20. Quadratic Equations h This is the graph of a parabola h = 10t – 2t2 (a) From the graph, what are the roots of the quadratic eqn. Both 8 h = 10t – 2t2 (b) What is the value of h for t = 1 and t = 4 t t = 0 t = 5 (c) What are the solutions of the quadratic equation 10t – 2t2 = 0 t = 0 and t = 5 (d) What is the solution of the quadratic equation 10t – 2t2 = 12.5 2.5

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

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

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

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

  25. Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following Common Factor Common Factor 16t – 6t2 = 0 x2 – 4x = 0 4t(8 – 3t) = 0 x(x – 4) = 0 x - 4 = 0 4t = 0 and 8 – 3t = 0 x = 0 and www.mathsrevision.com x = 4 t = 0 and t = 8/3 Created by Mr. Lafferty@www.mathsrevision.com

  26. Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following 100s2 – 25 = 0 x2 – 9 = 0 Difference 2 squares Difference 2 squares (10s – 5)(10s + 5) = 0 www.mathsrevision.com (x – 3)(x + 3) = 0 10s – 5 = 0 and 10s + 5 = 0 x = -3 x = 3 and s = 0.5 and s = - 0.5 Created by Mr. Lafferty@www.mathsrevision.com

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

  28. Solving Quadratic Equations Examples S4 Credit Common Factor 2x2 – 8 = 0 80 – 125e2 = 0 Common Factor 2(x2 – 4) = 0 5(16 – 25e2) = 0 Difference 2 squares Difference 2 squares www.mathsrevision.com 5(4 – 5e)(4 + 5e) = 0 2(x – 2)(x + 2) = 0 (4 – 5e)(4 + 5e) = 0 (x – 2)(x + 2) = 0 4 – 5e = 0 and 4 + 5t = 0 (x – 2) = 0 and (x + 2) = 0 x = 2 and x = - 2 t = 4/5 and t = - 4/5

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

  30. Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following x2 + 5x + 4 = 0 1 + x - 6x2 = 0 SAC Method SAC Method x 1 4 +3x www.mathsrevision.com x 1 1 -2x (x + 4)(x + 1) = 0 (1 + 3x)(1 – 2x) = 0 x + 4 = 0 and x + 1 = 0 1 + 3x = 0 and 1 - 2x = 0 x = - 4 and x = - 1 x = - 1/3 and x = 0.5

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

  32. Solving Quadratic Equations Examples S4 Credit Multiply out and rearrange Multiply out and rearrange Solve ( find the roots ) for the following (x + 4)2 =36 5x(2x + 1) - 10 = x(7x + 6) 3x2 - x - 10 = 0 x2 + 8x - 20 = 0 SAC Method SAC Method 3x +5 x 10 www.mathsrevision.com x - 2 x -2 (x + 10)(x - 2) = 0 (3x + 5)(x – 2) = 0 x + 10 = 0 and x - 2 = 0 3x + 5 = 0 and x - 2 = 0 x = - 10 and x = - 2 x = - 5/3 and x = 2

  33. Multiply through by 2(x - 1)(x + 2) to remove denominators Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following x - 4 x 1 2(x + 2) + 2(x – 1) = (x – 1)(x + 2) www.mathsrevision.com 2x + 4 + 2x – 2 = x2 + x - 2 (x - 4)(x + 1) = 0 x2 - 3x – 4 = 0 x - 4 = 0 and x + 1 = 0 SAC Method x = 4 and x = - 1

  34. Multiply through by x(x + 1) to remove denominators Solving Quadratic Equations Examples S4 Credit Solve ( find the roots ) for the following x 3 x - 2 6(x + 1) - 6x = x(x + 1) www.mathsrevision.com 6x + 6 – 6x = x2 + x (x + 3)(x - 2) = 0 x2 + x – 6 = 0 x + 3 = 0 and x - 2 = 0 SAC Method x = - 3 and x = 2

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

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

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

  38. Real-life Problems A rectangle garden is twice as long as it is wide. The area is 200m2. Find the dimensions of the rectangle garden. Let width be x Area = length x breadth 200 = 2xxx Length is 2x 200 = 2x2 x2= 100 x = 10 x = -10 and x must be positive ( We cannot get a negative length !!! ) Width is equal to 10m Length is equal to 20m

  39. Real-life Problems The height in metres of a rocket fired vertically upwards is give by the formula : h = 176t – 16t2 (a) When will the rocket be at a height of 160 metres. 160 = 176t – 16t2 16t2 - 176t + 160 = 0 t2 - 11t + 10 = 0 t = 10 and t = 1 (t– 10)(t – 1) = 0 (b) Is it possible for the rocket to h = 188 metres. Since 188 = 176t -16t2 has no solution not possible.

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

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

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

  43. Roots Formula S4 Credit Every quadratic equation can be rearranged into the standard form a, b and c are constants ax2 + bx + c = 0 Examples : find the constants a, b and c for the following www.mathsrevision.com 3x2 + x + 4 = 0 a = 3 b = 1 c = 4 x2 - x - 6 = 0 a = 1 b = -1 c = -6 x(x - 2) = 0 x2 – 2x = 0 a = 1 b = -2 c = 0 created by Mr. Lafferty

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

  45. Roots Formula S4 Credit Every quadratic equation can be rearranged into the standard form a, b and c are constants ax2 + bx + c = 0 In this form we can using the quadratic root formula to find the roots. www.mathsrevision.com created by Mr. Lafferty

  46. Roots Formula S4 Credit Example : Solve x2 + 3x - 3 ax2 + bx + c 1 3 -3 www.mathsrevision.com created by Mr. Lafferty

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

  48. Roots Formula Use the quadratic formula to solve the following : 2x2 + 4x + 1 = 0 x2 + 3x – 2 = 0 x = -1.7, -0.3 x = -3.6, 0.6 3x2 - 3x – 5 = 0 5x2 - 9x + 3 = 0 x = 1.4, 0.4 x = 1.9, -0.9 created by Mr. Lafferty

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

  50. Exam Type Questions

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