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Which ones are you using?

Learn how to accurately solve quadratic equations by completing the square and enhance your mathematical skills. Use PLT skills like creative thinking and independent inquiry.

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Which ones are you using?

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  1. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Real life cross/curricular links? Where are we in our journey? Which ones are you using? PLT Skills LESSON OBJECTIVES Always aim high! We are learning to: • Enhance Mathematical basic skills knowledge. (Which PLT skills?) • Accurately solve quadratic equations by completing the square. (Grades A-A*). AUTHOR www.mistrymaths.co.uk

  2. Which ones are you using? BRAIN IN GEAR Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills EXAMPLE DITDIONA can be rearranged to make ADDITION TASK Work out the following Mathematical anagrams: Quadratic Coefficient Rearrange EXTENSION Develop your own Mathematical anagrams as above as a creative thinker.

  3. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager STARTER PLT Skills TASK Distance = 925miles Speed = 74mph Time = ……….minutes 2) 3) 1) Write in standard form: 57.6 4cm 750 1 = 5.76 x 10 8cm 925 = = 12.5hours 74 N 15cm Area shaded black? Write the bearing. Solve: 2x + 5y = 15 3x - 2y = 13 5) 4) Area of circle – Area of triangle Π x r x r - 075° = - 3.14 x 7.5 x 7.5 = = - 176.625 16 4x + 10y = 30 6) 15x - 10y = 65 + 8m 2 160.6cm 95 = 19x 8m Π x d 360 3.14 x 16 360 b x h 8 x 4 x 5 = Work out the perimeter 10 6 2 2 4 4 10 + 5y = 15 + r + r + 8 + 8 5y = 5 = EXTENSION 1 y = 28.56m = = 60° Hexagon → Find the sum of one exterior angle in an regular decagon and one exterior angle in a regular hexagon? = 36° Decagon →

  4. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXAMPLE 1 Rewrite the following in the form (x a) b 2 2 2 x + 6x - 11 x - 8x + 5 (a) (b) b = 6 c = -11 b = -8 c = 5 + + - - Write down the formula first Write down the formula first b b b b 2 2 2 2 (x + ) – ( ) + c (x + ) – ( ) + c 2 2 2 2 ( ) ( ) 2 b 2 b Substitute in the values for b and c Substitute in the values for b and c c - ax + bx + c x + 2 + = 6 -8 6 2 2 2 -8 2 - 11 – 5 x – x 2 ) ) ( ( ) ) ( ( 2 + + + + 2 2 2 2 2 2 2 2 - 11 – – + 5 ) ( x ) ) ( x ) ( 3 3 ( 4 -4 + - 2 2 – - 11 x ) – + 5 x ) ( 3 9 ( 4 16 + - 2 2 x ) - 20 - x ) - 11 ( 3 ( 4 +

  5. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 1 (GRADE A) 2 Rewrite the following in the form (x a) b 2 2 2 2 2 2 2 2 3) 4) x - 8x + 11 x - 20x + 15 x + 2x - 9 x + 4x - 2 x - 14x - 12 x - 6x + 5 x + 12x - 7 x + 10x + 6 2) 1) + + - - 7) 5) 6) 8)

  6. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXAMPLE 2 Solve by completing the square. Leave your answer in the form a b 2 2 x - 6x – 3 = 0 x - 6x – 3 = 0 2 = 0 - x ) - 12 ( 3 + + + - - - b = -6 c = -3 2 - x ) ( 3 = 12 Write down the formula first - x 3 b b 12 = 2 2 (x + ) – ( ) + c = 0 ( ) ( ) 2 b 2 b c - 2 2 ax + bx + c x + 2 + = 12 x 3 = 2 2 Substitute in the values for b and c -6 -6 2 2 - 3 – x = 0 ) ) ( ( + + 2 2 2 2 – ( ) x ) - 3 ( 3 -3 - = 0 2 – x ) - ( 3 9 - 3 = 0 2 4 3 3 2 = 0 - x ) - 12 ( 3

  7. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 2 (GRADE A*) Solve the following by completing the square. Leave your answers in surd form. 4) 3) 2 2 2 2 2 2 2 x + 2x – 5 = 0 x - 4x - 2 = 0 x + 8x - 11 = 0 x + 10x + 7 = 0 x - 2x - 7 = 0 x + 12x + 3 = 0 x - 6x + 4 = 0 x - 14x – 4 = 0 2) 2 1) 7) 8) 6) 5)

  8. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXAMPLE 3 Solve by completing the square. Give your answer to 2 decimal places. 2 2 x + 2x – 9 = 0 x + 2x – 9 = 0 2 = 0 x ) - 10 ( 1 + + + - - b = 2 c = -9 2 + 1 x ) ( = 10 Write down the formula first + x 1 b b 10 = 2 2 (x + ) – ( ) + c = 0 ( ) ( ) 2 b 2 b c - 2 2 ax + bx + c x + 2 + = x 10 -1 = 2 2 Substitute in the values for b and c 2 2 2 2 - 9 – x = 0 ) ) ( ( + + x x x x + - 10 10 -1 2.16 -1 -4.16 = = = = 2 2 2 2 – ( ) x ) - 9 ( 1 1 + = 0 2 – x ) ( 1 1 - 9 + = 0 2 = 0 + x ) - 10 ( 1

  9. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 3 (GRADE A*) Solve the following by completing the square. Round your answers to 2d.p. where appropriate. 4) 3) 2 2 2 2 2 2 2 x + 2x – 3 = 0 x - 6x - 13 = 0 x + 2x - 9 = 0 x + 10x + 12 = 0 x - 2x - 21 = 0 x + 10x + 15 = 0 x - 16x – 13 = 0 x - 4x + 2 = 0 2) 2 1) 7) 8) 6) 5)

  10. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXTENSION 1 (GRADE A*) Solve the following by completing the square. Leave your answers in surd form. 2 2 2 2 2 2 3) 3x - 12x + 9 = 0 5x + 30x = 45 x - 5x = -3 2x + 4x – 8 = 0 2x + 12x – 20 = 0 x + 3x – 2 = 0 2) 1) Divide by 2 throughout Divide by 3 throughout Divide by 2 throughout 5) 6) 7) Make RHS = 0 Divide by 5 and change RHS = 0

  11. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXTENSION 2 (GRADE A*) Rearrange the following statements to give the complete solution, using the method of completing the square to the equation: 2 ax + bx + c = 0 G, F, A, I C, D, H, B, J, E,

  12. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO Draw a 3 x 3 grid as shown below: Place any nine answers below into the grid (repeats forbidden) -8 -3 D 6 -4.5 4 -2 A 2 B -9 -10 C 7 -50 3

  13. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x + 4x + 5 = 0 (x + ?) + 1 = 0 Find the value of ? below: 2

  14. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x + 14x + 48 = 0 (x + ?) - 1 = 0 Find the value of ? below: 7

  15. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 6x - 16 = 0 (x + ?) - 25 = 0 Find the value of ? below: -3

  16. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 x + 4x + 3 = 0 Which is answer is correct in surd form: + + + + - - - - A) x = -2 1 B) x = 2 1 C) x = -2 7 D) x = -2 7

  17. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 20x - 44 = 0 (x + ?) - 144 = 0 Find the value of ? below: -10

  18. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 16x + 64 = 0 (x + ?) = 0 Find the value of ? below: -8

  19. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 x - 8x + 7 = 0 Which is answer is correct in surd form: + + + + - - - - B) x = -4 9 A) x = -4 23 C) x = 4 9 D) x = 4 23

  20. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 (x + ?) + 40.25 = 0 x - 9x + 20 = 0 Find the value of ? below: -4.5

  21. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 8x + 18 = 0 (x - 4) + ? = 0 Find the value of ? below: -2

  22. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x + 4x + 7 = 0 (x + 2) + ? = 0 Find the value of ? below: 3

  23. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 x - 4x - 12 = 0 Which is answer is correct in surd form: + + + + - - - - B) x = -2 8 A) x = 2 8 C) x = -2 16 D) x = 2 16

  24. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x + 6x + 15 = 0 (x + 3) + ? = 0 Find the value of ? below: 6

  25. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x + 12x + 27 = 0 (x + 6) + ? = 0 Find the value of ? below: -9

  26. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 18x + 31 = 0 (x - 9) + ? = 0 Find the value of ? below: -50

  27. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 x + 10x + 24 = 0 Which is answer is correct in surd form: + + + + - - - - B) x = 5 1 A) x = -5 -1 C) x = -5 1 D) x = -5 49

  28. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 1 – BINGO 2 2 x - 12x + 40 = 0 (x - 6) + ? = 0 Find the value of ? below: 4

  29. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 2 – ASSESSING UNDERSTANDING 2 Bogdar rewrites the expression x + px + q by completing the square. He correctly does this and gets (x – 7) – 52. What are the values of p and q? 2 (x – 7)(x – 7) – 52 = 2 (x – 7) – 52 2 2 - 7x - 52 - 7x x x + 49 = - 14x - 3 = p q p = -14 and q = -3

  30. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager • LINK BACK TO OBJECTIVES • Accurately solve quadratic equations by completing the square. What grade are we working at? DISCOVERY PLT Skills

  31. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? COMPLETING THE SQUARE PLENARY ACTIVITY 1 – CONNECT 4 Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose.

  32. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? COMPLETING THE SQUARE PLENARY ACTIVITY 1 – CONNECT 4 ANSWERS Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose. {7, -9} {-1} {4.83, -6.83} {-5.45, -0.55} {11.08, -1.08} {7.16, 0.84} {-2, 0} {4, -2} {-2, -10} {6.32, -0.32} {-1.28, -11.72} {11.52, -2.52} {-3, -1} {5, 3} {3, -1} {4.41, 1.59} {5.56, -7.56} {-8.36, 0.36} {9, -1} {-2, 6} {-1, 5} {9.16, 2.84} {-5.61, 1.61} {3.45, -1.45} {8, -6} {12, -4} {-5.41, -2.59} {11.57, 0.43} {5.32, -1.32} {9.24, 0.76} {-3, -7} {-7, 1} {-3.73, -0.27} {1.27, -17.27} {-6.74, -0.74} {-3.41, -0.59}

  33. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? COMPLETING THE SQUARE PLENARY ACTIVITY 2 – GRADE A* EXAM QUESTION 1) (a) Find the values of a and b such that: 2 2 x + 6x – 3 = 0 x + 6x – 3 = (x + a) + b b = 6 c = -3 b b 2 2 (x + ) – ( ) + c = 0 2 – x ) + ( 3 9 - 3 = 0 2 2 6 6 2 2 - 3 – x = 0 ) ) ( ( + + 2 = 0 + x ) - 12 ( 3 2 2 2 2 2 – ( ) x ) - 3 ( 3 3 + = 0 2 = 0 x ) - 12 ( 3 + a b + + + + a = 3 and b = -12 - - - - (2 marks) (b) Hence, or otherwise, solve the equation: giving your answers in surd form. 12 2 = 0 + x ) - 12 ( 3 3 x 4 -3 = 2 + 3 x ) ( = 12 x 3 2 -3 = + x 3 = (3 marks) x 12 -3 =

  34. What grade are we working at? Where are we in our journey? What have you learnt? Draw your brain In your brain, write or draw everything you can remember about solving quadratic equations by completing the square. It can be a skill or a reflection, or something else that might be prominent in your brain.

  35. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity How well do you understand the task? . I fully understand I don’t understand I nearly understand www.mistrymaths.co.uk

  36. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity WWW (What Went Well) EBI (Even Better If) On your post it notes… Think about how you can improve your work. www.mistrymaths.co.uk

  37. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 1 (GRADE A) 2 Rewrite the following in the form (x a) b 2 2 2 2 2 2 2 2 3) 4) x - 8x + 11 x - 20x + 15 x + 2x - 9 x + 4x - 2 x - 14x - 12 x - 6x + 5 x + 12x - 7 x + 10x + 6 2) 1) + + - - 7) 5) 6) 8)

  38. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 2 (GRADE A*) Solve the following by completing the square. Leave your answers in surd form. 4) 3) 2 2 2 2 2 2 2 x + 2x – 5 = 0 x - 4x - 2 = 0 x + 8x - 11 = 0 x + 10x + 7 = 0 x - 2x - 7 = 0 x + 12x + 3 = 0 x - 6x + 4 = 0 x - 14x – 4 = 0 2) 2 1) 7) 8) 6) 5)

  39. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE TASK 3 (GRADE A*) Solve the following by completing the square. Round your answers to 2d.p. where appropriate. 4) 3) 2 2 2 2 2 2 2 x + 2x – 3 = 0 x - 6x - 13 = 0 x + 2x - 9 = 0 x + 10x + 12 = 0 x - 2x - 21 = 0 x + 10x + 15 = 0 x - 16x – 13 = 0 x - 4x + 2 = 0 2) 2 1) 7) 8) 6) 5)

  40. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXTENSION 1 (GRADE A*) Solve the following by completing the square. Leave your answers in surd form. 2 2 2 2 2 2 3) x - 5x = -3 2x + 4x – 8 = 0 3x - 12x + 9 = 0 5x + 30x = 45 2x + 12x – 20 = 0 x + 3x – 2 = 0 2) 1) 5) 6) 7)

  41. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE EXTENSION 2 (GRADE A*) Rearrange the following statements to give the complete solution, using the method of completing the square to the equation: 2 ax + bx + c = 0

  42. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills COMPLETING THE SQUARE MINI-PLENARY 2 – ASSESSING UNDERSTANDING 2 Bogdar rewrites the expression x + px + q by completing the square. He correctly does this and gets (x – 7) – 52. What are the values of p and q? 2

  43. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? COMPLETING THE SQUARE PLENARY ACTIVITY 1 – CONNECT 4 Connect 4 counters in a horizontal, vertical or diagonal line by correctly completing the square of the sum you choose.

  44. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? COMPLETING THE SQUARE PLENARY ACTIVITY 2 – GRADE A* EXAM QUESTION 1) (a) Find the values of a and b such that: 2 2 x + 6x – 3 = (x + a) + b x + 6x – 3 = 0 2 (2 marks) (b) Hence, or otherwise, solve the equation: giving your answers in surd form. (3 marks)

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