Lesson 20
Hart Interactive – Algebra 1
M4
ALGEBRA I
Exit Ticket Sample Solutions
Complete the square: 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒙𝒙 + 𝟑𝟑.
𝟏𝟏
𝟏𝟏 𝟐𝟐
𝟏𝟏
𝒂𝒂 �𝒙𝒙 + 𝒙𝒙� + 𝟑𝟑 → 𝒂𝒂 �𝒙𝒙 + � + 𝟑𝟑 −
𝒂𝒂
𝟐𝟐𝟐𝟐
𝟒𝟒𝟒𝟒
𝟐𝟐
Homework Problem Set Sample Solutions
S.147
For each equation below, complete the square and then identify the vertex of the quadratic function.
1. y = 3x2 – 24x – 1
2. y = 5x2 + 20x + 7
3. y = 4x2 – 12x + 9
y = 3(x2 – 8x) – 1
y = 5(x2 + 4x) + 7
y = 4(x2 – 6x) + 9
Y = 3(x2 – 8x + _) – 1 - __
Y = 5(x2 + 4x + _) + 7 - __
Y = 4(x2 – 6x + _) + 9 - __
Y = 3(x2 – 8x + 16) – 8 – 48
Y = 5(x2 + 4x + 4) + 7 – 20
Y = 4(x2 – 6x + 9) + 9 – 36
Y = 3(x – 4)2 – 56
Y = 5(x + 2)2 – 13
Y = 4(x – 3)2 – 27
Vertex: ( 4, -56 )
Vertex: ( -2, -13 )
4. y = 7x2 – 14x + 10
5. y = 3x2 – 6x – 4
Vertex: ( 3, -27 )
6. y = 2x2 + 8x + 1
y = 7(x2 – 2x) + 10
y = 3(x2 – 2x) – 4
y = 2(x2 + 4x) + 1
Y = 7(x2 – 2x + _) + 10 - __
Y = 3(x2 – 2x + _) – 4 - __
Y = 2(x2 + 4x + _) + 1 - __
Y = 7(x2 – 2x + 4) + 10 – 28
Y = 3(x2 – 2x + 4) – 4 – 12
Y = 2(x2 + 4x + 4) + 1 – 8
Y = 7(x – 2)2 – 18
Y = 3(x – 2)2 – 16
Y = 2(x + 2)2 – 7
Vertex: ( 2, -18 )
Lesson 20:
Unit 12:
Vertex: ( 2, -16 )
Complicated Quadratics
Completing the Square & The Quadratic Formula
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M4-TE-1.3.0-09.2015
Vertex: ( -2, -7 )
272
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 20
Hart Interactive – Algebra 1
M4
ALGEBRA I
7. y = 5x2 – 10x + 15
8. y = 2x2 – 12x – 13
9. y = x2 + 8x + 19
y = 5(x2 – 2x) + 15
y = 2(x2 – 6x) – 13
y = (x2 + 8x) + 1 9
Y = 5(x2 – 2x + _) + 15 - __
Y = 2(x2 – 6x + _) – 13 - __
Y = (x2 + 8x + _) + 19 - __
Y = 5(x2 – 2x + 1) + 15 – 5
Y = 2(x2 – 6x + 9) – 13 – 9
Y = (x2 + 8x + 16) + 19 – 16
Y = 5(x – 1)2 + 10
Y = 2(x – 3)2 – 22
Y = (x + 4)2 + 3
Vertex: ( 1, 10 )
Vertex: ( -2, -22 )
Vertex: (-4, 3 )
S.148
CHALLENGE PROBLEMS
Rewrite each expression by completing the square.
10. −𝟐𝟐𝒙𝒙𝟐𝟐 + 𝟖𝟖𝒙𝒙 + 𝟓𝟓
−𝟐𝟐�𝒙𝒙𝟐𝟐 − 𝟒𝟒𝟒𝟒 + 𝟒𝟒� + 𝟓𝟓 + 𝟖𝟖 → −𝟐𝟐(𝒙𝒙 − 𝟐𝟐)𝟐𝟐 + 𝟏𝟏𝟏𝟏
11. 𝟐𝟐. 𝟓𝟓𝒌𝒌𝟐𝟐 − 𝟕𝟕. 𝟓𝟓𝒌𝒌 + 𝟏𝟏. 𝟐𝟐𝟐𝟐
𝟒𝟒
𝟑𝟑
𝟐𝟐. 𝟓𝟓�𝒌𝒌𝟐𝟐 − 𝟑𝟑𝟑𝟑 + 𝟐𝟐. 𝟐𝟐𝟐𝟐� + 𝟏𝟏. 𝟐𝟐𝟐𝟐 − 𝟓𝟓. 𝟔𝟔𝟔𝟔𝟔𝟔 → 𝟐𝟐. 𝟓𝟓(𝒌𝒌 − 𝟏𝟏. 𝟓𝟓)𝟐𝟐 − 𝟒𝟒. 𝟑𝟑𝟑𝟑𝟑𝟑
12. 𝒃𝒃𝟐𝟐 + 𝟔𝟔𝒃𝒃 − 𝟓𝟓
𝟖𝟖𝟖𝟖
𝟐𝟐𝟐𝟐 𝟒𝟒
𝟗𝟗 𝟐𝟐 𝟒𝟒𝟒𝟒
𝟒𝟒 𝟐𝟐 𝟗𝟗
�𝒃𝒃 + 𝒃𝒃 + � − 𝟓𝟓 −
→ �𝒃𝒃 + � −
𝟐𝟐
𝟏𝟏𝟏𝟏
𝟒𝟒
𝟑𝟑
𝟒𝟒
𝟒𝟒
𝟑𝟑
13. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝒄𝒄𝟐𝟐 − 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏�𝒄𝒄𝟐𝟐 − 𝟏𝟏. 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟓𝟓𝟐𝟐 � + 𝟔𝟔𝟔𝟔𝟔𝟔 − 𝟑𝟑𝟑𝟑𝟑𝟑. 𝟔𝟔𝟔𝟔𝟔𝟔 → 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏(𝒄𝒄 − 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔)𝟐𝟐 + 𝟑𝟑𝟑𝟑𝟑𝟑. 𝟑𝟑𝟑𝟑𝟑𝟑
14. 𝟖𝟖𝒏𝒏𝟐𝟐 + 𝟐𝟐𝟐𝟐 + 𝟓𝟓
𝟏𝟏
𝟏𝟏
𝟏𝟏
𝟏𝟏 𝟐𝟐
𝟕𝟕
𝟖𝟖 �𝒏𝒏𝟐𝟐 + 𝒏𝒏 + � + 𝟓𝟓 − → 𝟖𝟖 �𝒏𝒏 + � + 𝟒𝟒
𝟒𝟒
𝟔𝟔𝟔𝟔
𝟖𝟖
𝟖𝟖
𝟖𝟖
Lesson 20:
Unit 12:
Complicated Quadratics
Completing the Square & The Quadratic Formula
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG I-M4-TE-1.3.0-09.2015
273
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.