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# 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