AA Section 6-5

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Completing the Square

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  • AA Section 6-5

    1. 1. Section 6-5 Completing the Square
    2. 2. How do we find the vertex of a standard form quadratic?
    3. 3. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4)
    4. 4. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 y + 4 = 3( x + 2x + 1)
    5. 5. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 y + 4 = 3( x + 2x + 1) 2 y + 4 = 3x + 6 x + 3
    6. 6. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 y + 4 = 3( x + 2x + 1) 2 y + 4 = 3x + 6 x + 3 2 y = 3x + 6 x − 1
    7. 7. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 2 y + 4 = 3( x + 2x + 1) y − 1= 2( x − 8 x + 16) 2 y + 4 = 3x + 6 x + 3 2 y = 3x + 6 x − 1
    8. 8. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 2 y + 4 = 3( x + 2x + 1) y − 1= 2( x − 8 x + 16) 2 2 y + 4 = 3x + 6 x + 3 y − 1= 2x − 16 x + 32 2 y = 3x + 6 x − 1
    9. 9. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 2 y + 4 = 3( x + 2x + 1) y − 1= 2( x − 8 x + 16) 2 2 y + 4 = 3x + 6 x + 3 y − 1= 2x − 16 x + 32 2 2 y = 3x + 6 x − 1 y = 2x − 16 x + 33
    10. 10. Warm-up Rewrite in standard form. 2 2 1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 4) 2 2 y + 4 = 3( x + 2x + 1) y − 1= 2( x − 8 x + 16) 2 2 y + 4 = 3x + 6 x + 3 y − 1= 2x − 16 x + 32 2 2 y = 3x + 6 x − 1 y = 2x − 16 x + 33 Inquiry: Can we switch from standard form to vertex form?
    11. 11. The binomial square 2 ( x − h)
    12. 12. The binomial square 2 ( x − h) 2 2 = x − 2hx + h
    13. 13. The binomial square 2 ( x − h) 2 2 = x − 2hx + h This is a perfect square trinomial, as it is the result of the binomial square.
    14. 14. Theorem (Completing the Square)
    15. 15. Theorem (Completing the Square) To complete the square 2 + bx, add (1/2b)2 x
    16. 16. Theorem (Completing the Square) To complete the square 2 + bx, add (1/2b)2 x 2 2 ( x + 1) = x + 2x + 1
    17. 17. Theorem (Completing the Square) To complete the square 2 + bx, add (1/2b)2 x 2 2 2 2 ( x + 1) = x + 2x + 1 ( x − 4) = x − 8 x + 16
    18. 18. Theorem (Completing the Square) To complete the square 2 + bx, add (1/2b)2 x 2 2 2 2 ( x + 1) = x + 2x + 1 ( x − 4) = x − 8 x + 16 2 1= (1/ 2 × 2)
    19. 19. Theorem (Completing the Square) To complete the square 2 + bx, add (1/2b)2 x 2 2 2 2 ( x + 1) = x + 2x + 1 ( x − 4) = x − 8 x + 16 2 2 1= (1/ 2 × 2) 16 = (1/ 2 × −8)
    20. 20. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90
    21. 21. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90
    22. 22. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x
    23. 23. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 ( b) 1 2
    24. 24. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 ()( ) × 18 1 1 b= 2 2
    25. 25. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) 1 1 2 2
    26. 26. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2
    27. 27. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2 2 y − 90 =x + 18 x + 81
    28. 28. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2 2 y − 90 + 81=x + 18 x + 81
    29. 29. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2 2 y − 90 + 81=x + 18 x + 81 2 y − 9=x + 18 x + 81
    30. 30. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2 2 y − 90 + 81=x + 18 x + 81 2 y − 9=x + 18 x + 81 2 y − 9=(x+9)
    31. 31. Example 1 Rewrite y = x2 +18x + 90 in vertex form. 2 y = x + 18 x + 90 -90 -90 2 y − 90 = x + 18 x 2 2 2 ( b) = ( ) × 18 = (9) = 81 1 1 2 2 2 y − 90 + 81=x + 18 x + 81 2 y − 9=x + 18 x + 81 2 y − 9=(x+9) 2 y=(x+9) + 9
    32. 32. How to complete the square
    33. 33. How to complete the square 1. Isolate the x terms
    34. 34. How to complete the square 1. Isolate the x terms 2. Make sure a = 1, then find b
    35. 35. How to complete the square 1. Isolate the x terms 2. Make sure a = 1, then find b 3. Add (1/2b)2 to both sides of the equation ***Be mindful that a = 1***
    36. 36. How to complete the square 1. Isolate the x terms 2. Make sure a = 1, then find b 3. Add (1/2b)2 to both sides of the equation ***Be mindful that a = 1*** 4. Factor the perfect square trinomial and simplify
    37. 37. How to complete the square 1. Isolate the x terms 2. Make sure a = 1, then find b 3. Add (1/2b)2 to both sides of the equation ***Be mindful that a = 1*** 4. Factor the perfect square trinomial and simplify 5. Check your answer
    38. 38. Example 2 Rewrite y = x2 -11x + 4 in vertex form.
    39. 39. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4
    40. 40. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x
    41. 41. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 ( ) 1 ×b 2
    42. 42. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 2 ( )( ) 1 1 ×b = × −11 2 2
    43. 43. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 2 2 ( )( )() 1 1 ×b = −11 × −11 = 2 2 2
    44. 44. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 2 2 121 ( )( ) ( )= 1 1 ×b = −11 × −11 = 4 2 2 2
    45. 45. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 2 2 121 ( )( ) ( )= 1 1 ×b = −11 × −11 = 4 2 2 2 2 y −4+ = x − 11x + 121 121 4 4
    46. 46. Example 2 Rewrite y = x2 -11x + 4 in vertex form. 2 y = x − 11x + 4 2 y − 4 = x − 11x 2 2 2 121 ( )( ) ( )= 1 1 ×b = −11 × −11 = 4 2 2 2 2 y −4+ = x − 11x + 121 121 4 4 2 ( ) 105 = x − 11 y+ 4 2
    47. 47. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form.
    48. 48. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1
    49. 49. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x
    50. 50. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x)
    51. 51. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 ( ) 1 ×b 2
    52. 52. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 ( ) =( ) 1 1 ×b × −4 2 2
    53. 53. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 2 ( ) =( )() 1 1 ×b × −4 = −2 2 2
    54. 54. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 2 =4 ( ) =( )() 1 1 ×b × −4 = −2 2 2
    55. 55. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 2 =4 ( ) =( )() 1 1 ×b × −4 = −2 2 2 2 y − 1+ 3(4) = 3( x − 4 x + 4)
    56. 56. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 2 =4 ( ) =( )() 1 1 ×b × −4 = −2 2 2 2 y − 1+ 3(4) = 3( x − 4 x + 4) 2 y − 1+ 12 = 3( x − 4 x + 4)
    57. 57. Example 3 Rewrite y = 3x2 -12x + 1 in vertex form. 2 y = 3x − 12x + 1 2 y − 1= 3x − 12x 2 y − 1= 3( x − 4 x) 2 2 2 =4 ( ) =( )() 1 1 ×b × −4 = −2 2 2 2 y − 1+ 3(4) = 3( x − 4 x + 4) 2 y − 1+ 12 = 3( x − 4 x + 4) 2 y + 11= 3( x − 2)
    58. 58. Notice: When a ≠ 1, we need to factor out the coefficient from both terms with an x.
    59. 59. Try these on your own or at the board. 2 2 1. y = 2x + 8 x + 5 2. y = 4 x + 36 x + 14 2 2 3. y = x − 10 x + 10 4. y = 5 x + 70 x − 9
    60. 60. Try these on your own or at the board. 2 2 1. y = 2x + 8 x + 5 2. y = 4 x + 36 x + 14 2 y + 3 = 2( x + 2) 2 2 3. y = x − 10 x + 10 4. y = 5 x + 70 x − 9
    61. 61. Try these on your own or at the board. 2 2 1. y = 2x + 8 x + 5 2. y = 4 x + 36 x + 14 2 ( ) 2 y + 3 = 2( x + 2) y + 67 = 4 x + 9 2 2 2 3. y = x − 10 x + 10 4. y = 5 x + 70 x − 9
    62. 62. Try these on your own or at the board. 2 2 1. y = 2x + 8 x + 5 2. y = 4 x + 36 x + 14 2 ( ) 2 y + 3 = 2( x + 2) y + 67 = 4 x + 9 2 2 2 3. y = x − 10 x + 10 4. y = 5 x + 70 x − 9 2 y − 15 = ( x − 5)
    63. 63. Try these on your own or at the board. 2 2 1. y = 2x + 8 x + 5 2. y = 4 x + 36 x + 14 2 ( ) 2 y + 3 = 2( x + 2) y + 67 = 4 x + 9 2 2 2 3. y = x − 10 x + 10 4. y = 5 x + 70 x − 9 2 2 y − 15 = ( x − 5) y + 254 = 5( x + 7)
    64. 64. Homework
    65. 65. Homework p. 374 #1 - 11 “You must dare to disassociate yourself from those who would delay your journey...Leave, depart, if not physically, then mentally. Go your own way, quietly, undramatically, and venture toward trueness at last.” - Vernon Howard

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