1. The document provides steps for completing the square of a quadratic equation to convert it from standard form to vertex form. It shows how to isolate the x terms, find the value of b, add (1/2b)2 to both sides, and factor to obtain the vertex form. 2. An example problem walks through rewriting y = x2 + 18x + 90 in vertex form by following the steps: isolating x terms, finding b, adding (1/2b)2, and factoring the perfect square trinomial to obtain the vertex form y - 9 = (x + 9)2. 3. Completing the square is a method to convert a quadratic equation from standard form

1. Section 6-5
Completing the Square

2. How do we find the vertex of a standard form
quadratic?

3. Warm-up
Rewrite in standard form.
2 2
1. y + 4 = 3( x + 1) 2. y − 1= 2( x − 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. 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. 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. 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. 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. 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. 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. The binomial square
2
( x − h)

12. The binomial square
2
( x − h)
2 2
= x − 2hx + h

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. Theorem (Completing the Square)

15. Theorem (Completing the Square)
To complete the square 2 + bx, add (1/2b)2
x

16. Theorem (Completing the Square)
To complete the square 2 + bx, add (1/2b)2
x
2 2
( x + 1) = x + 2x + 1

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. 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. 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. Example 1
Rewrite y = x2 +18x + 90 in vertex form.
2
y = x + 18 x + 90

21. Example 1
Rewrite y = x2 +18x + 90 in vertex form.
2
y = x + 18 x + 90
-90 -90

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. How to complete the square

33. How to complete the square
1. Isolate the x terms

34. How to complete the square
1. Isolate the x terms
2. Make sure a = 1, then find b

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. 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. 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. Example 2
Rewrite y = x2 -11x + 4 in vertex form.

39. Example 2
Rewrite y = x2 -11x + 4 in vertex form.
2
y = x − 11x + 4

40. Example 2
Rewrite y = x2 -11x + 4 in vertex form.
2
y = x − 11x + 4
2
y − 4 = x − 11x

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. 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. 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. 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. 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. 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. Example 3
Rewrite y = 3x2 -12x + 1 in vertex form.

48. Example 3
Rewrite y = 3x2 -12x + 1 in vertex form.
2
y = 3x − 12x + 1

49. Example 3
Rewrite y = 3x2 -12x + 1 in vertex form.
2
y = 3x − 12x + 1
2
y − 1= 3x − 12x

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. 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. 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. 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. 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. 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. 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. 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. Notice: When a ≠ 1, we need to factor out the
coefficient from both terms with an x.

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. 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. 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. 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. 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. Homework

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