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Complete Unit 12
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HighSchoolMathTeachers.com©2020
Table of Contents
Unit 12 Pacing Chart -------------------------------------------------------------------------------------------- 1
Algebra 1 Unit 12 Skills List ---------------------------------------------------------------------------------------- 5
Unit 12 Lesson Plans -------------------------------------------------------------------------------------------- 6
Day 146 Bellringer -------------------------------------------------------------------------------------------- 29
Day 146 Activity -------------------------------------------------------------------------------------------- 32
Day 146 Practice -------------------------------------------------------------------------------------------- 34
Day 146 Exit Slip -------------------------------------------------------------------------------------------- 37
Day 147 Bellringer -------------------------------------------------------------------------------------------- 39
Day 147 Activity -------------------------------------------------------------------------------------------- 42
Day 147 Practice -------------------------------------------------------------------------------------------- 44
Day 147 Exit Slip -------------------------------------------------------------------------------------------- 47
Day 148 Bellringer -------------------------------------------------------------------------------------------- 49
Day 148 Activity -------------------------------------------------------------------------------------------- 52
Day 148 Practice -------------------------------------------------------------------------------------------- 68
Day 148 Exit Slip -------------------------------------------------------------------------------------------- 70
Day 149 Bellringer -------------------------------------------------------------------------------------------- 72
Day 149 Activity -------------------------------------------------------------------------------------------- 75
Day 149 Practice -------------------------------------------------------------------------------------------- 77
Day 149 Exit Slip -------------------------------------------------------------------------------------------- 81
Week 30 Assessment -------------------------------------------------------------------------------------------- 83
Day 151 Bellringer -------------------------------------------------------------------------------------------- 89
Day 151 Activity -------------------------------------------------------------------------------------------- 90
Day 151 Practice -------------------------------------------------------------------------------------------- 101
Day 151 Exit Slip -------------------------------------------------------------------------------------------- 114
Day 152 Bellringer -------------------------------------------------------------------------------------------- 116
Day 152 Activity -------------------------------------------------------------------------------------------- 118
Day 152 Practice -------------------------------------------------------------------------------------------- 121
Day 152 Exit Slip -------------------------------------------------------------------------------------------- 123
Day 153 Bellringer -------------------------------------------------------------------------------------------- 125
Day 153 Activity -------------------------------------------------------------------------------------------- 127
Day 153 Practice -------------------------------------------------------------------------------------------- 129
Day 153 Exit Slip -------------------------------------------------------------------------------------------- 134
Day 154 Bellringer -------------------------------------------------------------------------------------------- 136
Day 154 Activity -------------------------------------------------------------------------------------------- 138
Day 154 Practice -------------------------------------------------------------------------------------------- 141
Day 154 Exit Slip -------------------------------------------------------------------------------------------- 147
Week 31 Assessment -------------------------------------------------------------------------------------------- 149
Day 156 Bellringer -------------------------------------------------------------------------------------------- 154
Day 156 Activity -------------------------------------------------------------------------------------------- 156
Day 156 Practice -------------------------------------------------------------------------------------------- 158
Day 156 Exit Slip -------------------------------------------------------------------------------------------- 164
Day 157 Bellringer -------------------------------------------------------------------------------------------- 166
Day 157 Activity -------------------------------------------------------------------------------------------- 168
Day 157 Practice -------------------------------------------------------------------------------------------- 171
Day 157 Exit Slip -------------------------------------------------------------------------------------------- 176
Day 158 Bellringer -------------------------------------------------------------------------------------------- 178
Day 158 Activity -------------------------------------------------------------------------------------------- 180
Day 158 Practice -------------------------------------------------------------------------------------------- 183
Day 158 Exit Slip -------------------------------------------------------------------------------------------- 197
Day 159 Bellringer -------------------------------------------------------------------------------------------- 199
Day 159 Activity -------------------------------------------------------------------------------------------- 201
Day 159 Practice -------------------------------------------------------------------------------------------- 203
Day 159 Exit Slip -------------------------------------------------------------------------------------------- 208
Week 32 Assessment -------------------------------------------------------------------------------------------- 210
Unit 12 Test -------------------------------------------------------------------------------------------- 215
CCSS Algebra 1 Pacing Chart – Unit 12
HighSchoolMathTeachers © 2020 Page 1
Unit Week Day CCSS Standards Mathematical Practices Objective I Can Statements
12 – Solve Quadratic Functions
30 – Factoring
146 CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to factor a quadratic expression to find the zeros of a function.
I can factor a quadratic expression to find the zeros of a function.
12 – Solve Quadratic Functions
30 – Factoring
147
CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*
CCSS.MATH.PRACTICE.MP6 Attend to precision.
The student will be able to create quadratic equations and inequalities in one variable and use them to solve problems.
I can create quadratic equations and inequalities in one variable and use them to solve problems.
12 – Solve Quadratic Functions
30 – Factoring
148
CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
The student will be able to recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
I can recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
12 – Solve Quadratic Functions
30 – Factoring
149 CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
The student will be able to factor a quadratic expression to find the zeros of a function to solve problems.
I can factor a quadratic expression to find the zeros of a function to solve problems.
12 – Solve Quadratic Functions
30 – Factoring
150 Assessment Assessment Assessment Assessment
CCSS Algebra 1 Pacing Chart – Unit 12
HighSchoolMathTeachers © 2020 Page 2
12 – Solve Quadratic Functions
31 – Completing the Square
151
CCSS.MATH.CONTENT.HSA.SSE.B.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
I can complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
12 – Solve Quadratic Functions
31 – Completing the Square
152
CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
The student will be able to complete the square in a quadratic function to determine roots, extreme values, and symmetry.
I can complete the square in a quadratic function to determine roots, extreme values, and symmetry.
12 – Solve Quadratic Functions
31 – Completing the Square
153
CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
The student will be able to write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
I can write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
CCSS Algebra 1 Pacing Chart – Unit 12
HighSchoolMathTeachers © 2020 Page 3
12 – Solve Quadratic Functions
31 – Completing the Square
154
CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
The student will be able to derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax2 + bx + c = 0).
I can derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax2 + bx + c = 0).
12 – Solve Quadratic Functions
31 – Completing the Square
155 Assessment Assessment Assessment Assessment
12 – Solve Quadratic Functions
32 – Systems and Quadratic Formula
156
CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
The student will be able to use the quadratic formula to solve quadratic equations for the zeros.
I can use the quadratic formula to solve quadratic equations for the zeros.
12 – Solve Quadratic Functions
32 – Systems and Quadratic Formula
157
CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
The student will be able to recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
I can recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
CCSS Algebra 1 Pacing Chart – Unit 12
HighSchoolMathTeachers © 2020 Page 4
12 – Solve Quadratic Functions
32 – Systems and Quadratic Formula
158
CCSS.MATH.CONTENT.HSA.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
The student will be able to solve a system of equations exactly (with algebra) and approximately (with graphs).
I can solve a system of equations exactly (with algebra) and approximately (with graphs).
12 – Solve Quadratic Functions
32 – Systems and Quadratic Formula
159
CCSS.MATH.CONTENT.HSA.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
The student will be able to solve a system of equations exactly (with algebra) and approximately (with graphs) to solve problems.
I can solve a system of equations exactly (with algebra) and approximately (with graphs) to solve problems.
12 – Solve Quadratic Functions
32 – Systems and Quadratic Formula
160 Assessment Assessment Assessment Assessment
Algebra 1 Unit 12 Skills List Name ____________________________________
HighSchoolMathTeachers©2020 Page 5
Algebra 1 Unit 12 Skills List
Number Week Unit CCSS Skill
68 30 12 A.SSE.3a Factor quadratic equations
69 30 12 A.SSE.3a Solve quadratic equations by factoring
70 31 12 A.SSE.3b Complete the square
71 31 12 F.IF.8 Use completing the square to find maximum
and minimum values
72 32 12 A.REI.4 Use the quadratic formula to solve quadratic
equations
73 32 12 A.REI.4 Be able to identify which process is best to
solve a quadratic equation
74 32 12 A.REI.7 Solve a system of equations containing one
quadratic and one linear function
Algebra 1 Unit 12 Skills List Name ____________________________________
HighSchoolMathTeachers©2020 Page 6
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 7
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 30 – Factoring
Day: 146
Common Core State Standard: CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines.
Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
Objective: The student will be able to factor a quadratic expression to find the zeros of a function.
I can statement: I can factor a quadratic expression to find the zeros of a function.
Procedures: 1. Students will complete the Day 146 Bellringer. 2. Students will work with partners and complete the Day-146-Activity. 3. The Day-146-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-146-Exit Slip before leaving for the day. 5. Use the Day-146-Practice as individual practice or homework.
Materials: Day 146 Bellringer Day 146 Activitiy Day 146 Presentation Day 146 Exit Slip Day 146 Practice
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 8
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 30 – Factoring
Day: 147
Common Core State Standard: CCSS.MATH.CONTENT.HSA.CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.*
Mathematical Practice: CCSS.MATH.PRACTICE.MP6 Attend to precision.
Objective: The student will be able to create quadratic equations and inequalities in one variable and use them to solve problems.
I can statement: I can create quadratic equations and inequalities in one variable and use them to solve problems.
Procedures: 1. Students will complete the Day 147 – Bellringers- Solve quadratics by factoring . 2. Students will work with partners and complete the Day-147-Activity. 3. The Day-147-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-147-Exit Slip before leaving for the day. 5. Use the Day-147-Practice as individual practice or homework.
Materials: Day 147 – Bellringers- Solve quadratics by factoring Day 146 Activitiy Day 146 Presentation Day 146 Exit Slip Day 146 Practice
Accommodations/Special Circumstances:
Technology:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 9
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 10
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 30 – Factoring
Day: 148
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.
Mathematical Practice: CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
Objective: The student will be able to recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
I can statement: I can recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
Procedures: 1. Students will complete the Day 148 Bellringer. 2. Students will work with partners and complete the Day-148-Activity. 3. The Day-148-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-148-Exit Slip before leaving for the day. 5. Use the Day-148-Practice as individual practice or homework.
Materials: Day 148 Bellringer Day 148 Activitiy Day 148 Presentation Day 148 Exit Slip Day 148 Practice
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 11
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 12
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 30 – Factoring
Day: 149
Common Core State Standard: CCSS.MATH.CONTENT.HSA.SSE.B.3.A Factor a quadratic expression to reveal the zeros of the function it defines.
Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
Objective: The student will be able to factor a quadratic expression to find the zeros of a function to solve problems.
I can statement: I can factor a quadratic expression to find the zeros of a function to solve problems.
Procedures: 1. Students will complete the Day 149 Bellringer. 2. Students will work with partners and complete the Day-149-Activity. 3. The Day-149-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-149-Exit Slip before leaving for the day. 5. Use the Day-149-Practice as individual practice or homework.
Materials: Day 149 Bellringer Day 149 Activitiy Day 149 Presentation Day 149 Exit Slip Day 149 Practice
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 13
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 31 – Completing the Square
Day: 151
Common Core State Standard: CCSS.MATH.CONTENT.HSA.SSE.B.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
Objective: The student will be able to complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
I can statement: I can complete the square in a quadratic expression and use it to find the maximum or minimum value of a function.
Procedures: 1. Students will complete the Day 151 Bellringer. 2. Students will work with partners and complete the Day-151-Activity. 3. The Day-151-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-151-Exit slip – Visual Understanding of proofs before leaving for the day. 5. Use the Day-151-Practice – Visual Understanding of proofs as individual practice or homework.
Materials: Day 151 Bellringer Day 151 Activitiy Day 151 Presentation Day 151 Exit Slip Visual Understanding of proofs Day 151 Practice Visual Understanding of proofs
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 14
Accommodations/Special Circumstances:
Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 15
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 31 – Completing the Square
Day: 152
Common Core State Standard: CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Mathematical Practice: CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
Objective: The student will be able to complete the square in a quadratic function to determine roots, extreme values, and symmetry.
I can statement: I can complete the square in a quadratic function to determine roots, extreme values, and symmetry.
Procedures: 1. Students will complete the Day 152 – Bellringers-ompleting square to find minimum and maximum values. 2. Students will work with partners and complete the Day-152-Activity-Completing the square to find Maximum and Minimum values. 3. The Day-152-Presentation-Completing the square to find Maximum and Minimum values will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-152-Exit slip – Completing square to find minimum and maximum values before leaving for the day. 5. Use the Day-152-Practice as individual practice or homework.
Materials: Day 152 – Bellringers-ompleting square to find minimum and maximum values Day-152-Activity-Completing the square to find Maximum and Minimum values Day-152-Presentation-Completing the square to find Maximum and Minimum values Day-152-Exit slip – Completi
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 16
Accommodations/Special Circumstances:
Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 17
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 31 – Completing the Square
Day: 153
Common Core State Standard: CCSS.MATH.CONTENT.HSF.IF.C.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Objective: The student will be able to write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
I can statement: I can write a quadratic function in an equivalent appropriate form (i.e. standard form, vertex form, and intercept form) to highlight items of interest (zeros, extreme values, and symmetry).
Procedures: 1. Students will complete the Day 153 Bellringer. 2. Students will work with partners and complete the Day-153-Activity. 3. The Day-153-Presentation will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-153-Exit Slip before leaving for the day. 5. Use the Day-153-Practice as individual practice or homework.
Materials: Day 153 Bellringer Day 153 Activitiy Day 153 Presentation Day 153 Exit Slip Day 153 Practice
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 18
Accommodations/Special Circumstances:
Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 19
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 31 – Completing the Square
Day: 154
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.B.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.
Mathematical Practice: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Objective: The student will be able to derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax²+bx+c=0).
I can statement: I can derive the quadratic formula by completing the square from the standard form of a quadratic equation (ax²2 + bx + c = 0).
Procedures: 1. Students will complete the Day 154 – Bellringers- Summary quadratic formula. 2. Students will work with partners and complete the Day-154-Activity-Summary quadratic formula. 3. The Day-154-Presentation-Quadratic formula will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-154-Exit slip -Summary quadratic formula before leaving for the day. 5. Use the Day-154-Practice -Summary quadratic formula as individual practice or homework.
Materials: Day 154 – Bellringers- Summary quadratic formula Day-154-Activity-Summary quadratic formula Day-154-Presentation-Quadratic formula Day-154-Exit slip -Summary quadratic formula Day-154-Practice -Summary quadratic formula
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 20
Accommodations/Special Circumstances:
Technology: Superscript: ⁻ ⁺ ⁼ ⁽ ⁾ ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ Subscript: ₊ ₋ ₌ ₍ ₎ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 21
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 31 – Completing the Square
Day: 155
Common Core State Standard: Assessment
Mathematical Practice: Assessment
Objective: Assessment
I can statement: Assessment
Procedures: Assessment
Materials: Weekly Assessment Unit 12
Accommodations/Special Circumstances:
Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 22
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 32 – Systems and Quadratic Formula
Day: 156
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Objective: The student will be able to use the quadratic formula to solve quadratic equations for the zeros.
I can statement: I can use the quadratic formula to solve quadratic equations for the zeros.
Procedures: 1. Students will complete the Day 156 – Bellringers- Driving the quadratic formula. 2. Students will work with partners and complete the Day-156-Activity-Driving the quadratic formula . 3. The Day-156-Presentation-Driving the quadratic formula will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-156-Exit slip -Driving the quadratic formula before leaving for the day. 5. Use the Day-156-Practice -Driving the quadratic formula as individual practice or homework.
Materials: Day 156 – Bellringers- Driving the quadratic formula Day-156-Activity-Driving the quadratic formula Day-156-Presentation-Driving the quadratic formula Day-156-Exit slip -Driving the quadratic formula Day-156-Practice -Driving the quadratic formula
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 23
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 24
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 32 – Systems and Quadratic Formula
Day: 157
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.B.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic
Mathematical Practice: CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
Objective: The student will be able to recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
I can statement: I can recognize the appropriate method to solve a quadratic equation: by inspection, taking the square root, completing the square, using the quadratic formula, and factoring.
Procedures: 1. Students will complete the Day 157 – Bellringers- Quadratic application practice. 2. Students will work with partners and complete the Day-157-Activity-Quadratic application practice. 3. The Day-157-Presentation-Quadratic application practice will be used to look for misconceptions and encourage discussion. 4. Students will complete Day-157-Exit slip -Quadratic application practice before leaving for the day.
Materials: Day 157 – Bellringers- Quadratic application practice Day-157-Activity-Quadratic application practice Day-157-Presentation-Quadratic application practice Day-157-Exit Slip -Quadratic application practice Day-157-Practice -Quadratic application practice
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 25
5. Use the Day-157-Practice -Quadratic application practice as individual practice or homework.
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 26
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 32 – Systems and Quadratic Formula
Day: 158
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
Mathematical Practice: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Objective: The student will be able to solve a system of equations exactly (with algebra) and approximately (with graphs).
I can statement: I can solve a system of equations exactly (with algebra) and approximately (with graphs).
Procedures: 1. Students will complete the Day 158 – Bellringers- Systems of linear and quadratic equations (Day 1). 2. Students will work with partners and complete the Day 158 – Activity- Systems of linear and quadratic equations (Day 1). 3. The Day 158 – Presentation- Systems of linear and quadratic equations (Day 1) will be used to look for misconceptions and encourage discussion. 4. Students will complete Day 158 – Exit Slip- Systems of linear and quadratic equations (Day 1) before leaving for the day.
Materials: Day 158 – Bellringers- Systems of linear and quadratic equations (Day 1) Day 158 – Activity- Systems of linear and quadratic equations (Day 1) Day 158 – Presentation- Systems of linear and quadratic equations (Day 1) Day 158 – Exit Slip- Systems of linear
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 27
5. Use the Day 158 –Practice- Systems of linear and quadratic equations (Day 1) as individual practice or homework.
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 28
Unit 12 – Solve Quadratic Functions
Course: Algebra 1
Topic: 32 – Systems and Quadratic Formula
Day: 159
Common Core State Standard: CCSS.MATH.CONTENT.HSA.REI.C.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.
Mathematical Practice: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
Objective: The student will be able to solve a system of equations exactly (with algebra) and approximately (with graphs) to solve problems.
I can statement: I can solve a system of equations exactly (with algebra) and approximately (with graphs) to solve problems.
Procedures: 1. Students will complete the Day 159 – Bellringers- Systems of linear and quadratic equations (Day 2). 2. Students will work with partners and complete the Day 159 – Activity- Systems of linear and quadratic equations (Day 2). 3. The Day 159 – Presentation- Systems of linear and quadratic equations (Day 2) will be used to look for misconceptions and encourage discussion. 4. Students will complete Day 159 – Exit Slip- Systems of linear and quadratic equations (Day 2) before leaving for the day.
Materials: Day 159 – Bellringers- Systems of linear and quadratic equations (Day 2) Day 159 – Activity- Systems of linear and quadratic equations (Day 2) Day 159 – Presentation- Systems of linear and quadratic equations (Day 2) Day 159 – Exit Slip- Systems of linear
Unit 12 Lesson Plans Name ____________________________________
HighSchoolMathTeachers©2020 Page 29
5. Use the Day 152 –Practice- Systems of linear and quadratic equations (Day 2) as individual practice or homework.
Accommodations/Special Circumstances: Technology:
Reflection:
Extra/Additional Resources:
Day 146 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 30
1. Find the square of the following 3, 5, −12 𝑎𝑛𝑑 25
2. Determine the square root of 25, 16, 81, 121
3. Express 225, 169 and 200 as a product of two equal numbers
Day 146 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 31
Answer Key
Day 146
1. 32 = 9, 52 = 25, (−12)2 = 144 , 252 = 625
2. √25 = ±5, ; √16 = ±4, √81 = ±9, √121 = ±11
3. 225 = 15 × 15, 169 = 13 × 13, 200 = 20 × 20
Day 146 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 32
Algebraic Method
The exact length of time takes for an object to fall to the ground can be found by algebraic method.
Using the height formula ℎ(𝑡) = −16𝑡2 + 𝑘, where k represents the initial height in feet and t is in
seconds, the exact solutions are ±√𝑘
16, although the negative root is not considered a meaningful
solution. Find the exact length of time it takes for an object to fall to the ground from each height below.
a. 144 feet
b. 180 feet
c. 200 feet
d. 95 feet
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Answer Key
1. a. 3 seconds
b. √11.25 𝑠𝑒𝑐𝑜𝑛𝑑𝑠;
c. √12.5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠;
d. √5.9375 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
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Find each square root. Round answers to the nearest hundredth.
1. √1
2. √81
3. √16
4. √225
5. √30
6. √18
7. √110
8. √55
Solve each equation. Round answers to the nearest hundredth.
9. 𝑥2 = 16
10. 𝑥2 = 900
11. 𝑥2 = 75
12. 𝑥2 =1
4
13. 𝑥2 =4
49
14. 𝑥2 =9
25
15. (𝑥 − 2)2 = 16
16. (𝑥 + 2)2 = 16
17. 𝑥2 − 4 = 0
18. 4 = (𝑥 + 3)2
19. −(𝑥 + 3)2 + 4 = 0
20. (𝑥 − 2)2 − 36 = 0
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21. (𝑥 + 6)2 = 3 22. (𝑥 − 2)2 = 14
Find the vertex, axis of symmetry, and zeros of each function. Sketch a graph
23. 𝑓(𝑥) = (𝑥 − 1)2 − 1 24. 𝑔(𝑥) = (𝑥 + 2)2 − 4
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Answer Key
1. 1 2. 9 3. 4 4. 15 5. 5.48 6. 4.24 7. 10.49 8. 7.42 9. ±4 10. ±30 11. ±8.66
12. ±1
2
13. ±2
7
14. ±3
5
15. −2, 6 16. −6, 2 17. ±2 18. −5, −1 19. −5, −1 20. −4, 8 21. −7.73, −4.27 22. −1.74, 5.74 23. Vertex: (1, −1); axis of symmetry: 𝑥 = 1; zeros: 0 and 2
24. Vertex: (−2, −4); axis of symmetry: 𝑥 = −2; zeros: 0 and -4
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Find each square root. If necessary, round to the nearest hundredth.
1. √169
2. √65
Solve each equation to the nearest hundredth.
3. 𝑥2 = 400
4. 𝑥2 =64
121
5. (𝑥 + 3)2 = 64
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Answer Key
1. 13
2. 8.06
3. ±20
4. ±8
11
5. −11, 5
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1. Expand the following (i). (4𝑥 + 2)(𝑥 − 3) (ii) 5𝑥(4 − 7𝑥)
(iii) (10𝑥 +1
2)(8𝑥 − 12)
2. Factor out the following (i). 3𝑥 + 9𝑥2 (ii). 9𝑥 − 12𝑥2
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Answer Key
Day 147
1. (i). 4𝑥2 − 10𝑥 − 6 (ii). 20𝑥 − 35𝑥2 (iii). 80𝑥2 − 116𝑥 − 6
2. (i). 3𝑥(1 + 3𝑥) (ii). 3𝑥(3 − 4𝑥)
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Like an integer, a monomial has prime factors.
1. Write the monomial whose prime factors are 2 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑥.
2. Write the monomial whose prime factors are 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥.
3. What is the greatest common factor of the monomials you wrote in Question 1 and 2?
4. What is the greatest common factor of 30𝑥4, 15𝑥2, and 10𝑥5?
5. Multiply each pair of binomials.
a. (𝑥 + 2)(𝑥 − 2)
b. (2𝑥 + 3)(2𝑥 − 3)
6. Describe the ways in which the products of Question 5 are similar.
7. Describe the ways in which the factors given in each part of Question 5 are related.
8. Multiply each pair of binomials.
a. (𝑥 + 2)(𝑥 + 2)
b. (2𝑥 − 3)(2𝑥 − 3)
9. Describe the ways in which the products of Question 8 are similar.
10. Describe the ways in which the factors given in which part of Question 8 are related.
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Answer Key
Day 147
1. 8𝑥2 2. 12𝑥5 3. 4𝑥2 4. 5𝑥2 5. a. 𝑥2 − 4
c. 4𝑥2 − 9 6. Each is the difference of two perfect squares 7. In both cases, the first terms of the factors are the same and the last terms are opposites. 8. A. 𝑥2 + 4𝑥 + 4
b. 4𝑥2 − 12𝑥 + 9 9. Each has a perfect square first term and a positive perfect square last term. The middle term is
double the product of the square roots of the first and last terms. 10. In both cases, the factors are identical.
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Factor. Check by multiplying.
1. 3𝑦2 − 6𝑦
2. 21𝑎3𝑏2 − 14𝑎2𝑏
3. 3𝑐(𝑟 − 𝑡) + 2𝑑(𝑟 − 𝑡)
Use factoring to evaluate the expression mentally.
4. 35 ∙ 49 + 35 ∙ 51
5. 22
7 ∙ 1600 −
22
7∙ 900
Factor.
6. 4ℎ2 − 25
7. 0.16𝑚2 − 0.25
8. (𝑎 + 𝑏)2 − 𝑐2
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9. WRITING MATHEMATICS Explain how the special product (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏2 can be used to mentally evaluate the product 25 ∙ 15. (Hint: Think of 25 as 20+5.)
10. MODELING Use Algeblocks to model the product 𝑥2 + 6𝑥 + 9, and find the factors of the trinomial.
Factor each trinomial. Check by multiplying.
11. 𝑥2 + 8𝑥 + 16
12. 𝑦2 + 10𝑦 + 25
13. 𝑧2 − 4𝑧 + 4
14. MODELING Use Algeblocks to model the product 𝑥2 − 𝑥 − 6, and find the factors of the trinomial.
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x
x
6
6
Factor each trinomial. Check by multiplying.
15. 𝑥2 + 4𝑥 + 3
16. 𝑎2 − 2𝑎 − 8
17. 𝑚2 + 𝑚 − 12
18. MODELING Use Algeblocks to model the product 2𝑥2 − 5𝑥 − 12, and find the factors of the trinomial.
19. GEOMETRY Express the shaded area as a polynomial in factored form.
Factor the trinomial. Check by multiplying.
20. 3𝑥2 + 7𝑥 + 2
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Answer Key
1. 3𝑦(𝑦 − 2) 2. 7𝑎2𝑏(3𝑎𝑏 − 2) 3. (𝑟 − 𝑡)(3𝑐 + 2𝑑) 4. 3500 5. 2200 6. (2ℎ + 5)(2ℎ − 5) 7. 0.01(4𝑚 + 5)(4𝑚 − 5) 8. (𝑎 + 𝑏 + 𝑐)(𝑎 + 𝑏 − 𝑐) 9. 25 × 15 = (20 + 5)(20 − 5) = 202 − 52 = 400 − 25 = 375 10. (𝑥 + 3)2: See additional Answer for mat.
11. (𝑥 + 4)2 12. (𝑦 + 5)2 13. (𝑧 − 2)2 14. (𝑥 + 2)(𝑥 − 3); See Additional Answer for mat 15. (𝑥 + 1)(𝑥 + 3) 16. (𝑎 + 2)(𝑎 − 4) 17. (𝑚 + 4)(𝑚 − 3) 18. (2𝑥 + 3)(𝑥 − 4); See Additional Answer for mat.
19. 𝑥2 − 36 = (𝑥 + 6)(𝑥 − 6) 20. (3𝑥 + 1)(𝑥 + 2)
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Factor. Check by multiplying.
1. 4𝑎2𝑏3 − 6𝑎𝑏5
2. 2𝑥(𝑥 − 2) − 5(𝑥 − 2)
3. 𝑥2 − 8𝑥 + 16
4. 𝑥2 − 121
5. 𝑥2 + 9𝑥 + 8
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Answer Key
1. 2𝑎𝑏3(2𝑎 − 3𝑏2) 2. (2𝑥 − 5)(𝑥 − 2) 3. (𝑥 − 4)2 4. (𝑥 + 11)(𝑥 − 11) 5. (𝑥 + 1)(𝑥 + 8)
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Factor each completely.
1. 𝑚2 + 9𝑚 + 8
2. 𝑎2 + 6𝑎 − 16
3. 𝑘2 − 18𝑘 + 80
4. 𝑛2 − 6𝑛 − 16
5. 𝑛2 + 3𝑛 − 28
6. 𝑥2 + 7𝑥 + 12
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Answer Key
Day 148
1. (𝑚 + 1)(𝑚 + 8) 2. (𝑎 + 8)(𝑎 − 2) 3. (𝑘 − 8)(𝑘 − 10) 4. (𝑛 + 2)(𝑛 − 8) 5. (𝑛 + 7)(𝑛 − 4) 6. (𝑥 + 3)(𝑥 + 4)
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The Box Method of Factoring a Polynomial.
Example: 10𝑥2 + 11𝑥 − 6
1st create a 2x2 box
2nd, in the top left corner put the first term and in the bottom right corner put the last term.
10𝑥2
−6
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3rd, multiply these two terms together to get −60𝑥2. Find two factors of −60𝑥2 that when
added together will give you the middle term 11𝑥. These are 15𝑥 and −4𝑥. Put these into the
open boxes.
−6
10𝑥2
15𝑥
−4𝑥
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4th, factor the terms in each row and in each column.
5th , the sum of the factors for the columns and the sum of the factors for the rows are the
polynomial’s factors: (2𝑥 + 3)(5𝑥 − 2)
−6
10𝑥2
15𝑥
−4𝑥
2𝑥 3
5𝑥
−2
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Explore 1: Factor 𝟑𝒙𝟐 + 𝟏𝟔𝒙 + 𝟓
Step 1 Place the first and last term in the box
Use the box model shown at the top of your activity worksheet to factor 3𝑥2 + 16𝑥 + 5. Place
the 𝑥2 term in the upper left square of the box. Place the constant term in the lower right
square of the box.
Step 2 List factors
Find the product of the terms in the box. Write it in the space provided on your worksheet.
Then list the factors of the product. Be sure to list the factors as the product of a number and 𝑥.
Step 3 Choose factors
Find the sum of the factors you found in step 2. Circle the factors that add up to the middle
term of 3𝑥2 + 16𝑥 + 5.
Step 4 Place the factors in the box
Place one of the factors you circled in step 3 in one of the empty squares. Place the other factor
in the remaining empty square.
Step 5 Find the greatest common factor
Find the GCF of the 1st column. Put this value in box (a).
Step 6 Use multiplication
The product of boxes (a) and (c) must equal the value in the upper left-hand square. To find the
value of (c) ask, “What do you multiply the value in box (a) by to get 3𝑥2?” Put your answer in
box (c).
Step 7 Fill in remain boxes
Repeat the procedure in Step 6 to find the values for boxes (b) and (d).
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Step 8 write the factors
The sum of boxes (a) and (b) form one of the factors. The sum of boxes (c) and (d) form the
other. Write the factors of the quadratic on your worksheet.
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DRAW CONCLUSIONS Use your observations to complete these exercises
1. Use the box model to factor 3𝑥2 + 13𝑥 + 12. You may want to refer to the steps in Explore 1.
In Exercises 2-4, use the box method to find the factors of the quadratic.
2. 3𝑥2 + 11𝑥 + 10 3. 4𝑥2 + 15𝑥 + 9 4. 2𝑥2 + 11𝑥 + 14
EXPLORE 2 Factor 𝟒𝒙𝟐 + 𝟓𝒙 − 𝟔 using the box method
STEP 1 Place the first and last terms in the box
Use the box model shown at the bottom of your activity worksheet to factor
4𝑥2 + 5𝑥 − 6. Place the 𝑥2 term in the upper left square of the box. Place the constant term in
the lower right square of the box. Find the product of the terms in the box and write it on your
worksheet.
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STEP 2 List factors
When the product of the first and last terms is negative, one of the factors is negative and one
of the factors is positive. For example, −13𝑥2 can have −1𝑥 and 13𝑥 as factors or 1𝑥 and
−13𝑥 as factors. List all possible factors of the product in the space provided on your
worksheet.
STEP 3 Choose factors
Find the sum of the factors you found in Step 2. Circle the factors that add up to the middle
term of 4𝑥2 + 5𝑥 − 6.
STEP 4 Place the factors in the box
Place one of the factors you circled in Step 3 in one of the empty squares. Place the other factor
in the remaining empty square.
STEP 5 Find the GCF and use multiplication
Find the GCF of the first column of the box. Put this value in box (a). Then use multiplication to
find the values that go in boxes (b) - (d). Remember to record whether the numbers in boxes
(b) – (d) are positive or negative.
STEP 6 Write the factors
The sum of boxes (a) and (b) form one of the factors of the quadratic. The sum of boxes (c) and
(d) form the other. Write the factors of the quadratic on your worksheet.
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DRAW CONCLUSIONS Use your observations to complete these exercises
5. Follow the steps in Explore 2 to complete the box model for 3𝑥2 − 19𝑥 + 6.
In Exercises 6-8, use the box method to find the factors of the quadratic.
6. 5𝑥2 − 8𝑥 − 4 7. 6𝑥2 + 5𝑥 − 4 8. 𝑥2 − 2𝑥 + 1
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Activity Worksheet
EXPLORE 1
3𝑥2 + 16 + 5
Product:___________Factors of the product:
3𝑥2 + 16𝑥 + 5=(___________________)(_________________)
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EXPLORE 2
4𝑥2 + 5𝑥 − 6
Product:_____________Factors of the product:
4𝑥2 + 5𝑥 − 6 =
(________________)(_____________)
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EXPLORE 1 WORKED-OUT SOLUTION SHEET
STEP 1
𝟑𝑿𝟐
5
STEP 2 Terms: 3𝑥2, 5 Product: 3𝑥2 ∙ 5 = 15𝑥2 Factors of 15𝑥2 1𝑥 ∙ 15𝑥 3𝑥 ∙ 5𝑥
STEP 3 Sum of
Factors: 1𝑥+15𝑥=16𝑥3𝑥+5𝑥=8𝑥
Since the sum of 1𝑥 and 15𝑥 is equal to the middle term, 16𝑥, circle those factors. Factors of
15𝑥2 (1𝑥∙15𝑥)3𝑥∙5𝑥
STEP 4 position of 𝟏𝒙 and 𝟏𝟓𝒙 may be switched
3𝑥2
1𝑥
15𝑥
5
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STEP 5 The GCF of 3𝑥2 and 15𝑥 is 3𝑥.
Step 6 STEP 6
STEP 7
STEP 8
𝟑𝒙𝟐 + 𝟏𝟔𝒙 + 𝟓 = (𝟑𝒙 + 𝟏)(𝒙 + 𝟓)
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EXPLORE 2 WORKED-OUT SOLUTION SHEET
STEP 1
𝟒𝒙𝟐
-6
STEP 2 Terms: 4𝑥2, −6 Product: 4𝑥2 ∙ −6 = −24𝑥2 Because the product is negative, remember that one factor will be negative and the other will be positive. Factors of −24𝑥2 1𝑥 ∙ −24𝑥 or −1𝑥 ∙ 24𝑥 2𝑥 ∙ −12𝑥 or −2𝑥 ∙ 12𝑥 3𝑥 ∙ −8𝑥 or −3𝑥 ∙ 8𝑥 4𝑥 ∙ −6𝑥 or −4𝑥 ∙ 6𝑥
STEP 3 Sum of factors of -24x:
1x + -24x = -23x
2x + -12x = -10x
3x + -8x = -5x
4x + -6x = -2x
-1x + 24x = 23x
-2x + 12x = 10x
-3x + 8x = 5x
- 4x + 6x = 2x
STEP 4 position of 8x and -3x may be switched
𝟒𝒙𝟐
𝟖𝒙
-3x
-6
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𝑥 2
4𝑥
−3
4𝑥2 8𝑥
−3𝑥 −6
STEP 5
STEP 6
𝟒𝒙𝟐 + 𝟓𝒙 − 𝟔 = (𝟒𝒙 + (−𝟑))(𝒙 + 𝟐)
= (𝟒𝒙 − 𝟑)(𝒙 + 𝟐)
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Answer Key
EXPLORE 1 AND 2
See worked-out solution sheets.
DRAW CONCLUSIONS
1. Position of x-terms may vary.
2. (3𝑥 + 5)(𝑥 + 2) 3. (𝑥 + 3)(4𝑥 + 3) 4. (2𝑥 + 7)(𝑥 + 2) 5. Position of x-terms may vary
In the diagram below:
(3𝑥 − 1)(𝑥 − 6)
6. (5𝑥 + 2)(𝑥 − 2) 7. (2𝑥 − 1)(3𝑥 + 4) 8. (𝑥 − 1)(𝑥 − 1)
(3𝑥 + 4)(𝑥 + 3)
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Teacher Notes
ACTIVITY PREPARATION AND MATERIALS
Each student will need an activity worksheet. Students may need extra paper to write their factors.
It will be helpful for students to have the worked-out solution sheets while doing Explores 1 and 2. Instruct the students to use the worked-out solutions to check their work at each step.
Review multiplying binomials using the box method before beginning this activity. This will help students feel comfortable using multiplication to find the factors.
ACTIVITY MANAGEMENT
Some students will be able to mentally figure out which two factors add to get the middle term. Encourage all students to write out the factors initially, especially when working with negative numbers.
Encourage students to check their answer using multiplication.
Common Error Students need to be careful when factoring the product of the diagonal. Make sure they account for negative signs when necessary. Remind students that a negative times a negative is positive.
A-level Alternative Work through the Explores as a class. Make sure you model the box method on the board for students. Fill in the box as you read through the steps. After completing Explore 1, you may want students to try Draw Conclusions Exercise 1 on their own or with a partner before continuing with Explore 2. After Explore 2, students can try Draw Conclusions Exercise 5.
C-level Alternative Do not provide students with the worked-out solutions.
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Activity and Closure Questions
Ask these question as a class
1. Describe the signs of the factors of a quadratic that has a negative middle term and positive last term, such as 𝑥2 − 5𝑥 + 6.
𝑨𝒏𝒔𝒘𝒆𝒓: Sample answer: The factors of a quadratic with a negative middle term and a
positive last term will both be negative. For example, 𝑥2 − 5𝑥 + 6 = (𝑥 − 3)(𝑥 − 2).
2. Describe the signs of the factors of a quadratic that has a negative middle term and a negative last term, such as 𝑥2 − 𝑥 − 6 = (𝑥 − 3)(𝑥 + 2)
LESSON TRANSITION
This activity provides students with a concrete method for factoring. You may want to do the
examples in the book using the box method instead of doing the activity. If you do the activity,
you may want to do Examples 4 and 5 before you assign homework.
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Factor each trinomial. Check by multiplying.
1. 𝟐𝒙𝟐 + 𝟓𝒙 + 𝟐
2. 𝟐𝒂𝟐 + 𝟕𝒂 + 𝟔
3. 𝟒𝒙𝟐 − 𝟒𝒙 − 𝟏𝟓
4. 𝟗 + 𝟗𝒙 + 𝟐𝒙𝟐
5. 𝟖 + 𝟏𝟒𝒓 + 𝟑𝒓𝟐
6. 𝟏𝟐 − 𝟏𝟑𝒂 − 𝟒𝒂𝟐
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Answer Key
1. (2𝑥 + 1)(𝑥 + 2)
2. (2𝑎 + 3)(𝑎 + 2)
3. (2𝑥 + 3)(2𝑥 − 5)
4. (3 + 2𝑥)(3 + 𝑥)
5. (2 + 3𝑟)(4 + 𝑟)
6. (4 + 𝑎)(3 − 4𝑎)
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Factor each completely.
1) 3𝑝2 − 2𝑝 − 5
2) 3𝑛2 − 8𝑛 + 4
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Answer Key
1) (𝟑𝒑 − 𝟓)(𝒑 + 𝟏) 2) (𝟑𝒏 − 𝟐)(𝒏 − 𝟐)
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Factor.
1) 15𝑛2 − 27𝑛 − 6
2) 5𝑥2 − 18𝑥 + 9
3) 4𝑛2 − 15𝑛 − 25
4) 4𝑥2 − 35𝑥 + 49
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Answer Key
1. 3(5𝑛 + 1)(𝑛 − 2)
2. (5𝑥 − 3)(𝑥 − 3)
3. (𝑛 − 5)(4𝑛 + 5)
4. (𝑥 − 7)(4𝑥 − 7)
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You know that solutions to the equation −𝑥2 − 6𝑥 − 5 = 0 can be found by
factoring and by using the Zero Product Property. Why are zeros of the functions
𝑓(𝑥) = −𝑥2 − 6𝑋 − 5 and the solutions to the given equation the same?
1. What are the factors of −𝑥2 − 6𝑥 − 5 = 0?_________________________
2. Find the x-intercepts of the function 𝑓(𝑥) = −𝑥2 − 6𝑥 − 5.
3. Write the expression −𝑥2 − 6𝑥 − 5 in the form (𝑥 − ℎ)2 + 𝑘 by completing the square.
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4. On the grid provided, graph the parabola in the form (𝑥 − ℎ)2 + ℎ and the lines that represent the factors you found in Exercise 1.
5. Find the x-intercept of each linear function.
6. Compare the x-intercepts of the parabola with the x-intercepts of each linear function. Describe the relationship.
7. How are the x-intercepts related to the zeros of a function?
8. Why are the zeros of the functions and the solutions to the corresponding quadratic equation the same?
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Answer Key
1. (𝑥 + 5)(𝑥 + 1) 2. −5 and −1 3. −(𝑥 + 3)2 + 4 4.
5. The x-intercept of 𝑦 = 𝑥 + 5 is 𝑥 = −5, and the x-intercept of 𝑦 = 𝑥 + 1 is 𝑥 = −1.
6. Each linear function intersects the parabola at an x-intercept. 7. The x-intercepts are the zeros of a function. 8. Each linear function intersects the quadratic function at an x-intercept.
Since the x-intercepts are the zeros of a function, the zeros of the function and the solutions by factoring are the same.
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Solve each equation by factoring.
1. (𝑘 + 1)(𝑘 − 5) = 0
2. (𝑎 + 1)(𝑎 + 2) = 0
3. (4𝑘 + 5)(𝑘 + 1) = 0
4. (2𝑚 + 3)(4𝑚 + 3) = 0
5. 𝑥2 − 11𝑥 + 19 = −5
6. 𝑛2 + 7𝑛 + 15 = 5
7. 𝑛2 − 10𝑛 + 22 = −2
8. 𝑛2 + 3𝑛 − 12 = 6
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9. 6𝑛2 − 18𝑛 − 18 = 6
10. 7𝑟2 − 14𝑟 = −7
11. 𝑛2 + 8𝑛 = −15
12. 5𝑟2 − 44𝑟 + 120 = −30 + 11𝑟
13. −14𝑘2 − 8𝑘 − 3 = −3 − 5𝑘2
14. 𝑏2 + 5𝑏 − 35 = 3𝑏
15. 3𝑟2 − 16𝑟 − 7 = 5
16. 6𝑏2 − 13𝑏 + 3 = −3
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17. 7𝑘2 − 6𝑘 + 3 = 3
18. 35𝑘2 − 22𝑘 + 7 = 4
19. 7𝑥2 + 2𝑥 = 0
20. 10𝑏2 = 27𝑏 − 18
21. 8𝑥2 + 21 = −59𝑥
22. 15𝑎2 − 3𝑎 = 3 − 7𝑎
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Answer Key
1. {−1, 5}
2. {−1, −2}
3. {−5
4, −1}
4. {−3
2, −
3
4}
5. {3, 8}
6. {−5, −2}
7. {6, 4}
8. {3, −6}
9. {4, −1}
10. {1}
11. {−5, −3}
12. {6, 5}
13. {−8/9 , 0}
14. {−7, 5}
15. {−2
3, 6}
16. {2
3,
3
2}
17. {6
7, 0}
18. {1
5,
3
7}
19. {−2
7, 0}
20. {6
5,
3
2}
21. {−3
8, −7}
22. {1
3, −
3
5}
Day 149 Exit Slip Name ____________________________________
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Solve the quadratics.
1. 7𝑣2 − 42 = −35𝑣
2. 𝑘2 = −4𝑘 − 4
Day 149 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 83
Answer Key
1. {−6, 1} 2. {−2}
84
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 30
HighSchoolMathTeachers©2020
85
Week 30
Weekly Assessments
86
Week #30 - Factoring
1. Factor each of the following quadratics.
a. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 5 b. 𝑓(𝑥) = 𝑥2 − 8𝑥 + 7
c. 𝑓(𝑥) = 2𝑥2 + 𝑥 − 6 d. 𝑓(𝑥) = 6𝑥2 + 10𝑥 − 24
2. Solve each of the quadratics by factoring:
a. 0 = 𝑥2 − 11𝑥 − 60 b. 0 = 𝑥2 + 17𝑥 + 72
c. −𝑥2 + 𝑥 + 20 = 0 d. 2𝑥2 + 15𝑥 + 7 = 0
87
Week 30 - KEYS
Weekly Assessments
88
Week #30 - Factoring Answer Key 1. Factor each of the following quadratics.
a. 𝑓(𝑥) = 𝑥2 + 4𝑥 − 5
(𝒙 + 𝟓)(𝒙 − 𝟏)
b. 𝑓(𝑥) = 𝑥2 − 8𝑥 + 7
(𝒙 − 𝟕)(𝒙 − 𝟏)
c. 𝑓(𝑥) = 2𝑥2 + 𝑥 − 6
(𝟐𝒙 − 𝟑)(𝒙 + 𝟐)
d. 𝑓(𝑥) = 6𝑥2 + 10𝑥 − 24
𝟐(𝟑𝒙 − 𝟒)(𝒙 + 𝟑)
2. Solve each of the quadratics by factoring:
a. 0 = 𝑥2 − 11𝑥 − 60
(𝒙 − 𝟏𝟓)(𝒙 + 𝟒) = 𝟎 𝒙 = 𝟏𝟓, 𝒙 = −𝟒
b. 0 = 𝑥2 + 17𝑥 + 72
(𝒙 + 𝟗)(𝒙 + 𝟖) = 𝟎 𝒙 = −𝟗, 𝒙 = −𝟖
c. −𝑥2 + 𝑥 + 20 = 0
(𝒙 + 𝟒)(𝒙 − 𝟓) = 𝟎 𝒙 = −𝟒, 𝒙 = 𝟓
d. 2𝑥2 + 15𝑥 + 7 = 0
(𝒙 + 𝟕)(𝟐𝒙 + 𝟏) = 𝟎 𝒙 = −𝟕, 𝒙 = −𝟎. 𝟓
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Factor each of the following perfect square trinomials.
1. 𝑎2 + 4𝑎 + 4
2. 49 + 14𝑎 + 𝑎2
3. 𝑦2 – 8𝑦 + 16
4. 49𝑥2 + 28𝑥𝑦 + 4𝑦2
5. 𝑟2 + 24𝑟 + 144
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Answer Key.
Day 151
1. (𝑎 + 2)(𝑎 + 2)
2. (7 + 𝑎)(7 + 𝑎)
3. (𝑦 – 4)(𝑦 – 4)
4. (7𝑥 + 2𝑦)(7𝑥 + 2𝑦)
5. (𝑟 + 12)(𝑟 + 12)
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1. Use your calculator to graph the equation: 𝑦 = 𝑥2 + 6𝑥 + 9. Write the equation in vertex form.
2. Sketch algebra tiles to model the equation𝑦 = 𝑥2 + 6𝑥 + 9.
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Recall the values of each of the algebra tiles. The value of the tile is its area. We will only be working
with positive (red tiles) representations of algebraic expressions.
Our original equation was written in standard form,𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. Since it is usually much easier to
graph a parabola if the equation is in vertex form,𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, often we try to rewrite the
equation from standard form into vertex form. We will use algebra tiles to help us understand why this
procedure works.
When we are trying to write an equation in vertex form, we need to have a perfect square to make the
(𝑥 − ℎ)2 part of the equation. When the quadratic equation we are given is not a perfect square, we
arrange the parts to form a perfect square, adding the unit squares we need or keeping track of any
extras we may have. This activity will help you discover how to start the process of forming the perfect
square from what you are given.
𝑥2𝑇𝑖𝑙𝑒
AREA = 𝑥 ∙ 𝑥 = 𝑥2𝑢𝑛𝑖𝑡𝑠
𝑥 𝑇𝑖𝑙𝑒
AREA = 1 ∙ 𝑥 = 𝑥 𝑢𝑛𝑖𝑡𝑠
𝑈𝑛𝑖𝑡 𝑇𝑖𝑙𝑒
AREA = 1 ∙ 1 = 1 𝑢𝑛𝑖𝑡𝑠
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3. Create a partial square with algebra tiles to represent 𝑥2 + 2𝑥 + ____________ . a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the completed square? L = W =
c) Replace c and ? with numbers to make the statement true;𝑥2 + 2𝑥 + 𝑐 = (𝑥+? )2.
4. Create a partial square with algebra tiles to represent 𝑥2 + 4𝑥 + ____________.
a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the completed square? L = W=
c) Replace c and ? with numbers to make the statement true: 𝑥2 − 4𝑥 + 𝑐 = (𝑥+? )2
5. Create a partial square with algebra tiles to represent 𝑥2 − 6𝑥 + ________________.
a) How many unit tiles do you need to complete the square?
b) What are the dimensions of the complete square? L= W=
c) Replace c and ? with numbers to make the statement true: 𝑥2 − 6𝑥 + 𝑐 = (𝑥+? )2
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6. What is the relationship between the coefficient of x in the standard form of the equation and the number of x’s you have down one side of your algebra tile diagram?
7. What is the relationship between the numbers of x’s down one side of the algebra tile diagram and
the question mark in your perfect square binomial? 8. What is the relationship between the coefficient of x in the standard form of the equation and the
question mark in your perfect square binomial? 9. What is the relationship between the question mark of your perfect square binomial and the
number of blocks (or units) you had to add to make your diagram a perfect square? 10. In the expression = 𝑥2 + 𝑏𝑥 + 𝑐 , how do you use b to get the value of c to form a perfect
square? Use the examples above (3-5) to explain your answer. 11. Try these problems—Fill in the missing “c” and then rewrite the trinomial as a perfect square
binomial.
a) 𝑥2 − 10𝑥 + 𝑐 b) 𝑥2 − 4𝑥 + 𝑐
c) 𝑥2 + 12𝑥 + 𝑐 d) 𝑥2 − 12𝑥 + 𝑐
e) 𝑥2 + 7𝑥 + 𝑐 f) 𝑥2 + 𝑏𝑥 + 𝑐
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Represent each expression by sketching algebra tiles. Try to create a square of tiles. When doing so keep
the following rules in mind:
• You may only use one 𝑥2 tile in each square.
• You must use all the 𝑥2 and x-tiles. Unit tiles are the only ones that can be leftover or borrowed.
• If you need more unit tiles to create a square you have to “borrow” them. The number you borrow will
be a negative quantity.
Standard form Number
of 𝑥2 Tiles
Number of x Tiles
Number of Unit
Tiles Sketch of the Square
Length of the Square
Area if the Square
(Length)2
Unit Tiles Left Over
(+) Borrowed
(-)
Expression Combining
Previous Two Columns
𝑥2 + 2𝑥 + 3 1 2 3
y 1
x
1
𝑥 + 1 (𝑥 + 1)2 2 (𝑥 + 1)2 + 2
𝑥2 + 4𝑥 + 1
𝑥2 + 6𝑥 + 10
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12. What is the name of the form for the combined expression in the last column?
Convert the following equations from standard form to vertex form by completing the square.
13. 𝑦 = 𝑥2 − 8𝑥 + 11 14. 𝑦 = 𝑥2 + 16𝑥 + 14
Solving quadratic equations by completing the square:
Steps: Example: 𝑥2 + 6𝑥 − 16 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0
a) Move constant term of quadratic to the other side. Write the equation in the form 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______
𝑥2 + 6𝑥 − 16 + 16 = 0 + 16 𝑥2 + 6𝑥 + _____ = 16 + _____
b) Complete the square by adding a constant to both sides.
𝑥2 + 6𝑥 + 9 = 16 + 9
c) Rewrite the left side of the equation as a binomial squared and simplify the right side.
(𝑥 + 3)2 = 25
d) Square root both sides. 𝑥 + 3 = ±5
e) Solve for X. 𝑥 + 3 = −5 𝑥 + 3 = 5 𝑥 + 3 − 3 = −5 − 3 𝑥 + 3 − 3 = 5 − 3 𝑥 = −8 𝑥 = 2
Solve the following equations by completing the square using the steps above.
15. 𝑥2 − 4𝑥 − 32 = 0 16. 𝑥2 + 8𝑥 + 7 = 0
a) a)
b) b)
c) c)
d) d)
e) e)
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If the leading coefficient ≠ 1, we must first divide each term by “a” so that the coefficient of the 𝑥2 term
is 1. Then complete the same steps above.
17. 2𝑥2 + 4𝑥 − 3 = 0 18. 3𝑥2 + 18𝑥 + 12 = 0
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Answer Key
1.
𝑦 = (𝑥 + 3)2 Vertex form
2.
3.
a) We need 1 unit tiles to complete the square. b) 𝐿 = 𝑥 + 1; 𝑊 = 𝑥 + 1 c) C=1 ?=1
𝑥2
x x X
X 1 1 1
X 1 1 1
X 1 1 1
𝑥2
x
X 1
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4.
a) We need 4 unit tiles to complete the square. b) 𝐿 = 𝑥 + 2 ; 𝑊 = 𝑥 + 2 c) C=4 ?=-2
5.
a) We need 9 unit tiles to complete the square. b) 𝐿 = 𝑥 + 3 ; 𝑊 = 𝑥 + 3 c) C=9 ?=-3
6. The number of x’s you have down one side is equal to half of the coefficient of x. 7. The numbers of x’s down one side of the algebra tile diagram is equal to the question mark in
your perfect square binomial. 8. The question mark in the perfect square binomial is equal to half of the coefficient of x. 9. The square of the question mark of the perfect square binomial is equal to the number of units
you have to add to make your diagram. 10. To get the value of c, take half of b and square it. 11. a) 𝑥2 − 10𝑥 + 25 = (𝑥 − 5)2
b) 𝑥2 − 4𝑥 + 4 = (𝑥 − 2)2 c) 𝑥2 + 12𝑥 + 36 = (𝑥 + 6)2 d) 𝑥2 − 12𝑥 + 36 = (𝑥 − 6)2
e) 𝑥2 + 7𝑥 +49
4= (𝑥 +
7
2)
f) 𝑥2 + 𝑏𝑥 +𝑏2
4= (𝑥 +
𝑏
2)
2
𝑥2
x X
X 1 1
X 1 1
𝑥2
x x x
X 1 1 1
X 1 1 1
X 1 1 1
Day 151 Activity Name ____________________________________
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Represent each expression by sketching algebra tiles.
Standard form Number
of 𝑥2 Tiles
Number of x Tiles
Number of Unit
Tiles Sketch of the Square
Length of the Square
Area if the Square
(Length)2
Unit Tiles Left Over
(+) Borrowed
(-)
Expression Combining
Previous Two Columns
𝑥2 + 2𝑥 + 3 1 2 3
y 1
x
1
𝑥 + 1 (𝑥 + 1)2 2 (𝑥 + 1)2 + 2
𝑥2 + 4𝑥 + 1 1 4 1
x 1 1
X
1
1
𝑥 + 2 (𝑥 + 2)2 −3 (𝑥 + 2)2 − 3
𝑥2 + 6𝑥 + 10 1 6 10
𝑥 + 3 (𝑥 + 3)2 1 (𝑥 + 3)2 + 1
12. the name of the form for the combined expression in the last column is a vertex form.
13. 𝑦 = (𝑥 − 4)2 − 5 14. 𝑦 = (𝑥 + 8)2 − 50
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Steps: Example: 𝑥2 + 6𝑥 − 16 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0
a) Move constant term of quadratic to the other side. Write the equation in the form 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______
𝑥2 + 6𝑥 − 16 + 16 = 0 + 16 𝑥2 + 6𝑥 + _____ = 16 + _____
𝑥2 + 8𝑥 − 20 + 20 = 0 + 20
b) Complete the square by adding a constant to both sides.
𝑥2 + 6𝑥 + 9 = 16 + 9 𝑥2 + 8𝑥 + 16 = 20 + 16
c) Rewrite the left side of the equation as a binomial squared and simplify the right side.
(𝑥 + 3)2 = 25 (𝑥 + 4)2 = 36
d) Square root both sides. 𝑥 + 3 = ±5 𝑥 + 4 = ±6
e) Solve for X. 𝑥 + 3 = −5 𝑥 + 3 = 5 𝑥 + 3 − 3 = −5 − 3 𝑥 + 3 − 3 = 5 − 3 𝑥 = −8 𝑥 = 2
𝑥 + 4 = −6 𝑥 + 4 = 6 𝑥 = −10 𝑥 = 2
15. 𝑥2 − 4𝑥 − 32 = 0
a) 𝑥2 − 4𝑥 + _____ = 32 + _____
b) 𝑥2 − 4𝑥 + 4 = 32 + 4
c) (𝑥 − 2)2 = 36
d) 𝑥 − 2 = ±6
e) 𝑥 − 2 = −6 𝑥 − 2 = 6
𝑥 = −4 𝑥 = 8
16. 𝑥2 + 8𝑥 + 7 = 0
a) 𝑥2 + 8𝑥 + _____ = −7 + ______
b) 𝑥2 + 8𝑥 + 16 = −7 + 16
c) (𝑥 + 4)2 = 9
d) 𝑥 + 4 = ±3
e) 𝑥 + 4 = −3 𝑥 + 4 = 3
𝑥 = −7 𝑥 = −1
17. 2𝑥2 + 4𝑥 − 3 = 0
𝑥2 + 2𝑥 −3
2= 0
a) 𝑥2 + 2𝑥 + _____ =3
2+ _____
b) 𝑥2 + 2𝑥 + 1 =3
2+ 1
c) (𝑥 + 1)2 =5
2
d) 𝑥 + 1 = ±√5
2
e) 𝑥 + 1 = √5
2 𝑥 + 1 = −√
5
2
𝑥 = √5
2− 1 𝑥 = −√
5
2− 1
18. 3𝑥2 + 18𝑥 + 12 = 0
𝑥2 + 6𝑥 + 4 = 0
a) 𝑥2 + 6𝑥 + _____ = −4 + _____
b) 𝑥2 + 6𝑥 + 9 = −4 + 9
c) (𝑥 + 3)2 = 5
d) 𝑥 + 3 = ±√5
e) 𝑥 + 3 = −√5 𝑥 + 3 = √5
𝑥 = −√5 − 3 𝑥 = √5 − 3
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Use the information below to answer questions 1 – 6. Using the relevant number of boxes, determine the constant term to complete the square.
1. 𝑥2 + 6𝑥
2.𝑥2 + 8𝑥
3. 𝑥2 − 2𝑥
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4. 𝑥2 − 10𝑥
5. 4𝑥2 + 4𝑥
6. 4𝑥2 − 4𝑥
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Use the information below to answer questions 7 – 13.
Complete the tiles to make a perfect square and write the expression representing the diagram.
Orange or red tiles implies negative and green or gray represents positive.
7.
8.
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9.
10.
11.
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12.
13.
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14.
Use the information below to answer questions 15 –17
Write the coefficient of 𝑥 of the quadratic equation if the tiles below are supposed to represent
a complete square.
15.
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HighSchoolMathTeachers©2020 Page 108
16.
17.
Use the information below to answer questions 18 –20
Write the constant of the quadratic equation if the tiles below are supposed to represent a
complete square.
18.
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19.
20.
Day 151 Practice Name ___________________________________
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Answer Keys Day 151:
1. 𝑥2 + 6𝑥 + 9
2. 𝑥2 + 8𝑥 + 16
3. 𝑥2 − 2𝑥 + 1
4. 𝑥2 − 10𝑥 + 25
5. 4𝑥2 + 4𝑥 + 1
6. 4𝑥2 − 4𝑥 + 1 7. 𝑥2 + 8𝑥 + 16
8. 𝑥2 − 12𝑥 + 36
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9. 9𝑥2 − 12𝑥 + 4
10. 9𝑥2 + 6𝑥 + 1
11. 𝑥2 + 4𝑥 + 4
Or 𝑥2 − 4𝑥 + 4
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12. 𝑥2 + 8𝑥 + 16
Or 𝑥2 − 8𝑥 + 16
13. 𝑥2 − 2𝑥 + 1
Or 𝑥2 + 2𝑥 + 1
Day 151 Practice Name ___________________________________
HighSchoolMathTeachers©2020 Page 113
14. 16𝑥2 − 16𝑥 + 4
15. 4
16. 6
17. 6
18. 4
19. 9
20. 9
Day 151 Exit Slip Name ____________________________________
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Draw color tiles representing the quadratic expression 𝑥2 + 4𝑥 and show how you would
complete the square.
Day 151 Exit Slip Name ____________________________________
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Answer Keys Day 151:
𝑥2 + 4𝑥 + 4
Day 152 Bellringer Name ____________________________________
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Express the following expressions in the form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
1. 𝑦 = 𝑥2 − 4𝑥 + 1
2. 𝑦 = 5 − 16𝑥 − 2𝑥2
3. 𝑦 = 3𝑥2 − 9𝑥 + 7
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Answer Key
Day 152
1. 𝑦 = 𝑥2 − 4𝑥 + 1 = (𝑥 − 2)2 − 3 2. 𝑦 = 5 − 16𝑥 − 2𝑥2 = −2(𝑥 + 4)2 + 37
3. 𝑦 = 3𝑥2 − 9𝑥 + 7 = 3 (𝑥 −3
2 )
2+
1
4
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Consider the following equation: 𝑓(𝑥) = 8𝑥 − 2𝑥2 − 3
1. Draw a table of values for 𝑥 and 𝑓(𝑥) for integer values in the range −3 ≤ 𝑥 ≤ 5.
𝑥
𝑓(𝑥)
2. Find the highest value of 𝑓(𝑥) from the above table.
3. Identify the value of 𝑥 for which the value in 2 occurs.
4. Express the equation above in the standard form of a quadratic function.
5. Enclose the first two terms in parentheses and factor appropriately so that the coefficient of
𝑥2 is 1.
6. Complete the square in the parentheses
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7. Express the result in 6 above so that you have an expression of the form 𝑎(𝑥 − ℎ)2 + 𝑘
where 𝑎 can be positive or negative.
8. Compare the value of 𝑘 and ℎ to the answers for 2 and 3 respectively.
9. Make a conclusion based on the results from 8 above.
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Answer Keys Day 152:
1.
𝑥 -3 -2 -1 0 1 2 3 4 5
𝑓(𝑥) -45 -27 -13 -3 3 5 3 -3 -13
2. 5
3. 2
4. 𝑓(𝑥) = −2𝑥2 + 8𝑥 − 3 5. 𝑓(𝑥) = −2(𝑥2 − 4𝑥) − 3 6. 𝑓(𝑥) = −2(𝑥2 − 4𝑥 + 22) + 5 7. 𝑓(𝑥) = −2(𝑥 − 2)2 + 5 8. 𝑘 = 5, ℎ = 2
The values are the same as the answers for questions 2 and 3. 9. When a quadratic equation is expressed in the form 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘 where 𝑎 < 0, the
value of 𝑘 is the maximum valueof 𝑓(𝑥) while the value of ℎ is the 𝑥 −coordinate of the point where the maximum value occurs.
Day 152 Practice Name ____________________________________
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Solve each equation by completing the square.
1. 𝑎2 + 2𝑎 − 3 = 0
2. 𝑎2 − 2𝑎 − 8 = 0
3. 𝑘2 + 8𝑘 + 12 = 0
4. 𝑎2 − 2𝑎 − 48 = 0
5. 𝑥2 + 12𝑥 + 20 = 0
6. 𝑘2 − 8𝑘 − 48 = 0
7. 𝑝2 + 2𝑝 − 63 = 0
8. 𝑝2 − 8𝑝 + 21 = 6
9. 𝑚2 + 10𝑚 + 14 = −7
10. 𝑣2 − 2𝑣 = 3
Day 152 Practice Name ____________________________________
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Answer Key
1. {1, −3}
2. {4, −2}
3. {−2, −6}
4. {8, −6}
5. {−2, −10}
6. {12, −4}
7. {7, −9}
8. {5, 3}
9. {−3, −7}
10. {3, −1}
Day 152 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 123
Determine the minimum or the maximum value of the following equation.
𝑦 = 1 − 3𝑥2 + 18𝑥
Day 152 Exit Slip Name ____________________________________
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Answer Keys Day 152:
𝑦 = −3(𝑥 − 3)2 + 28
The maximum value is 28
Day 153 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 125
1. Solve each equation for x:
(i). 2𝑥 + 𝑡 = 𝑝𝑥
(ii). 𝑥
5+ 𝑠 =
4
2𝑥
(iii). 𝑡𝑥
𝑚+ 𝑟𝑥 = 1
2. Complete the square and then factor:
(i). 4𝑥2 − 𝑥
(ii) 3𝑥2 + 2𝑥
(iii). 𝑝𝑥2 − 𝑟𝑥
Day 153 Bellringer Name ____________________________________
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Answer Key
Day 153
1. (i). 𝑥 =𝑡
𝑝−2
(ii). 𝑥 =5
9𝑠
(iii).𝑥 =𝑚
𝑡+𝑚𝑟
2. (i). 4 (𝑥 −1
8)
2−
1
16
(ii).3 (𝑥 +1
3)
2−
1
3
(iii).𝑝 (𝑥 −𝑟
2𝑝)
2−
𝑟2
4𝑝
Day 153 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 127
Consider the following equation 4𝑥2 + 8𝑟𝑥 + 𝑘 = 0
1. Add −𝑘 on both sides of equal sign
2. Factor 4 from the left hand side
3. Complete the square in the parenthesis in 2 above
4. Simplify the parenthesis so that you have an equation of the form 𝑎(𝑥2 + 𝑑𝑥 + 𝑒2) = 𝑓.
5. Factor the square in the bracket using appropriate method
6. Solve the equation for 𝑥.
Day 153 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 128
Answer Keys Day 153:
1. 4𝑥2 + 8𝑟𝑥 = −𝑘
2. 4(𝑥2 + 2𝑟𝑥) = −𝑘
3. 4(𝑥2 + 2𝑟𝑥 + 𝑟2 − 𝑟2) = −𝑘
4. 4(𝑥2 + 2𝑟𝑥 + 𝑟2) = 4𝑟2 − 𝑘
5. 4(𝑥 + 𝑟)2 = 4𝑟2 − 𝑘
6. 𝑥 =−2𝑟±√4𝑟2−𝑘
2
Day 153 Practice Name ____________________________________
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Solve each equation for 𝑥, simplifying where possible.
1. (𝑥 − 2𝑝)2 = 3
2.(𝑥 + 3𝑡)2 =3
5
3. (𝑥 − 4)2 = 5
4. (𝑥 + 3)2 = 1
5. (𝑥 −4
2𝑟)
2
= 𝑞
6. (𝑥 −4
25𝑝)
2
=1
9𝑡2
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7.(𝑥 − 𝑟)2 =3𝑟2
4
8. (𝑥 − 3𝑟)2 =9
25𝑡
9.(𝑥 + 4𝑤)2 = 4 +1
4𝑤
10.(𝑥 −3
4𝑡)
2
=𝑟
4
11. (𝑥 −3
4𝑠)
2
=𝑡
4+ 1
12.(𝑥 − 2𝑝)2 =1
2𝑡
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13. (𝑥 − 3𝑟)2 =1
4− 𝑡
14.(𝑥 − 3𝑑)2 =3
4+ 𝑟
15.(𝑥 −𝑟
5)
2
=𝑟
4
16. 𝑝 (𝑥 +4
9𝑞)
2
= 4𝑝 +𝑞
4
17.3 (𝑥 −5
9𝑡)
2
= 9
18. 2 (𝑥 −3
7𝑟)
2
= 4𝑟 + 3
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19. 4(𝑥 + 3𝑟)2 = 19
20. 𝑟2 (𝑥 −3
4𝑡)
2
= 𝑡2
Day 153 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 133
Answer Keys Day 153:
1. 𝑥 = 2𝑝 ± 1.732
2. 𝑥 = −3𝑡 ± 0.7746
3. 𝑥 = 6.236 𝑜𝑟 𝑥 = 1.764
4. 𝑥 = −4 𝑜𝑟 𝑥 = −2
5. 𝑥 = 2𝑟 ± √𝑞
6. 𝑥 =4
25𝑝 ±
𝑡
3
7. 𝑥 = 𝑟 (2±√3
2)
8. 𝑥 = 3𝑟 ±3
5√𝑡
9. 𝑥 = −4𝑤 ±√16+𝑤
2
10. 𝑥 =3
4𝑡 ±
√𝑟
2
11. 𝑥 =3
4𝑠 ±
√𝑡+4
2
12. 𝑥 = 2𝑝 ± √𝑡
2
13. 𝑥 = 3𝑟 ±√1−4𝑡
2
14. 𝑥 = 3𝑑 ±√3+4𝑟
2
15. 𝑥 =𝑟
5±
√𝑟
2
16. 𝑥 = −4
9𝑞 ±
1
2√16𝑝 + 𝑞
17. 𝑥 =5
9𝑡 ± 1.732
18. 𝑥 =3
7𝑟 ± √4𝑟 + 3
19. 𝑥 = −3𝑟 ± 2.179
20. 𝑥 = (3
4±
1
𝑟) 𝑡
Day 153 Exit Slip Name ____________________________________
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Express 𝑥 in terms of 𝑙 and 𝑚 if (𝑥 − 𝑙)2 = 𝑚
Day 153 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 135
Answer Keys Day 153:
𝑥 = 𝑙 ± √𝑚
Day 154 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 136
1. Solve for 𝑥 in the following questions:
(i). (𝑥 − 4)2 = 9𝑡
(ii). (2𝑥 + 𝑟)2 = 16
2. Find the coordinates of the vertex of the following: 𝑦 = 2𝑥2 − 24𝑥 + 9
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Answer Key
Day 154
1. (i) 𝑥 = 4 ± 3√𝑡
(ii). 𝑥 = −𝑟
2± 2
2. (6, −63)
Day 154 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 138
Consider the following expression: 8𝑥2 + 4𝑥 − 1
1. Group the terms so that the first and the second term are in one group and the constant
term is in its own group.
2. Complete the square of the expression in the parenthesis
3. Factor the expression in the parenthesis.
4. Express the result in 3 above in the form of 𝑦 = 𝑟(𝑥 − ℎ)2 + 𝑘 where 𝑟 ≠ 0, ℎ and 𝑘 are
constants.
Day 154 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 139
5. Identify the relative maximum or the minimum value of the function.
6. Identify the vertex of the function
7. Let 𝑦 = 0 in the function and determine the value of 𝑥.
Day 154 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 140
Answer Keys Day 154:
1. (8𝑥2 + 4𝑥) − 1
2. 8 (𝑥2 +1
2𝑥 + (
1
4)
2
) −1
2− 1
3. 8 (𝑥 +1
4)
2
−1
2− 1
4. 𝑦 = 8 (𝑥 +1
4)
2
−3
2
5. Relative minimum is −3
2
6. Vertex(−1
4, −
3
2 )
7. 8 (𝑥 +1
4)
2
−3
2= 0
𝑥 = −1
4±
√3
4
Day 154 Practice Name ____________________________________
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For questions 1-5, solve the equations for 𝑥, simplifying completely.
1. (𝑥 − 2𝑑)2 = 9
2.(𝑥 − 6𝑝)2 = 25𝑝2
3. (2𝑥 + 4)2 =25
36
4. (2𝑥 + 3𝑠)2 =1
𝑛2
5. (𝑥 −𝑟
2)
2
=𝑞2
4
Day 154 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 142
For questions 6 – 14, compare the square:
6. 𝑥2 − 8𝑥
7. 𝑥2 − 22𝑥
8. 4𝑥2 − 40𝑥
9. 81𝑥2 − 36𝑥
10. 4𝑥2 + 8𝑥
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11. 𝑝2𝑥2 + 2𝑝𝑥
12. 𝑎2𝑥2 − (4𝑝)2𝑥
13. 25𝑥2 +𝑎
2𝑠𝑥
14. 36𝑥2 − 4𝑟𝑥
Day 154 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 144
For questions 15 – 16, draw square tiles representing the completed square whose part is given
below. Use different colors for negative and positive coefficients of 𝑥.
15. 4𝑥2 − 4𝑥
16. 𝑥2 + 6𝑥
Day 154 Practice Name ____________________________________
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Determine the minimum or maximum of the following functions.
17. 𝑦 = 𝑥2 + 4𝑥 + 2
18. 𝑦 = 2𝑥2 − 9𝑥 + 3
19. 𝑦 = −4𝑥2 + 8𝑥 + 3
20.𝑦 = −3𝑥2 + 7𝑥 + 8
Day 154 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 146
Answer Keys Day 154:
1. 𝑥 = 2𝑑 ± 3
2. 𝑥 = 6𝑝 ± 5𝑝 or 11p and 1p
3. 𝑥 = −2 ±5
12 or -19/12 and
-29/12
4. 𝑥 = −3
2𝑠 ±
1
2𝑛
5. 𝑥 =𝑟±𝑞
2
6. 𝑥2 − 8𝑥 + 16
7. 𝑥2 − 22𝑥 + 121
8. 4𝑥2 − 40𝑥 + 100
9. 81𝑥2 − 36𝑥 + 4
10. 4𝑥2 + 8𝑥 + 4
11. 𝑝2𝑥2 + 2𝑝𝑥 + 1
12. 𝑎2𝑥2 − 16𝑝2𝑥 +64𝑝4
𝑎2
13. 25𝑥2 +𝑎
2𝑠𝑥 +
𝑎2
400𝑠2
14. 36𝑥2 − 4𝑟𝑥 +𝑟2
9
15.
16.
17. Minimum value is -2
18. Minimum value −7.125
19. Maximum value 7
20. Maximum value 49
12
Day 154 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 147
Find the minimum or the maximum value of 𝑦 = 2𝑥2 + 14𝑥 − 2
Day 154 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 148
Answer Keys Day 154:
𝑦 = 2 (𝑥 +7
2)
2
−53
2
The function has a minimum value of −53
2.
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 31
HighSchoolMathTeachers©2020
150
Week 31
Weekly Assessments
151 Week 31
Week #31 - Completing the Square
1. Rewrite each quadratic in vertex format by completing the square
a. 𝑓(𝑥) = 𝑥2 + 16𝑥 + 71 b. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 5
c. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 4) d. 𝑓(𝑥) = −9𝑥2 − 162𝑥 − 731
2. Complete the square to determine the minimum/maximum values for each quadratic.
a. 𝑓(𝑥) = 𝑥2 + 10𝑥 + 33 b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5
c. 𝑓(𝑥) = 𝑥2 + 4𝑥 d. 𝑓(𝑥) = 6𝑥2 + 12𝑥 + 13
152 Week 31 - KEYS
Week 31 - KEYS
Weekly Assessments
153 Week 31 - KEYS
Week #31 - Completing the Square Answer Key 1. Rewrite each quadratic in vertex format by completing the square
a. 𝑓(𝑥) = 𝑥2 + 16𝑥 + 71
𝒇(𝒙) = (𝒙 + 𝟖)𝟐 + 𝟕
b. 𝑓(𝑥) = 𝑥2 − 2𝑥 − 5
𝒇(𝒙) = (𝒙 − 𝟏)𝟐 − 𝟔
c. 𝑓(𝑥) = (𝑥 + 5)(𝑥 + 4)
𝒇(𝒙) = (𝒙 + 𝟒. 𝟓)𝟐 − 𝟎. 𝟐𝟓
d. 𝑓(𝑥) = −9𝑥2 − 162𝑥 − 731
𝒇(𝒙) = −𝟗(𝒙 + 𝟗)𝟐 − 𝟐
2. Complete the square to determine the minimum/maximum values for each quadratic.
a. 𝑓(𝑥) = 𝑥2 + 10𝑥 + 33
𝒇(𝒙) = (𝒙 + 𝟓)𝟐 + 𝟖 (−𝟓, 𝟖)
Minimum value is 8
b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5
𝒇(𝒙) = (𝒙 − 𝟑)𝟐 − 𝟒 (𝟑, −𝟒)
Minimum value is -4
c. 𝑓(𝑥) = 𝑥2 + 4𝑥
𝒇(𝒙) = (𝒙 + 𝟐)𝟐 − 𝟒 (−𝟐, −𝟒)
Minimum value is -4
d. 𝑓(𝑥) = 6𝑥2 + 12𝑥 + 13
𝒇(𝒙) = 𝟔(𝒙 + 𝟏)𝟐 + 𝟕 (−𝟏, 𝟕)
Minimum value is 7
Day 156 Bellringer Name ________________________________
HighSchoolMathTeachers©2020 Page 154
Day 156
1. Supply the constant needed to complete the square:
(i). 4𝑥2 − 8𝑥
(ii). (3𝑥)2 − 30𝑥
2. Factor the following expressions
(i). 𝑐𝑥2 + 4 − 4√𝑐𝑥
(ii). 81𝑥2 + 108𝑥 + 36
(iii). 25𝑥2 + 130𝑥 + 169
Day 156 Bellringer Name ________________________________
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Answer Key
Day 156
1. (i). 4𝑥2 − 8𝑥 + 4
(ii). (3𝑥)2 − 30𝑥 + 25
2. (i). (√𝑐𝑥 − 2)2
(ii). 9(3𝑥 + 2)2
(iii). (5𝑥 + 13)2
Day 156 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 156
Consider the following function 𝑓(𝑥) = 𝑥2 + 2𝑝𝑥 − 𝑟
1. Group the terms so that the first and the second term are in one group and the constant
term in its own group.
2. Equate the expression to zero and then keep the variable terms on the left side and move the
constant term to the right side.
3. Complete the square of the left side.
4. Factor on the left hand side.
5. Express 𝑥 in terms of 𝑝 and 𝑟.
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HighSchoolMathTeachers©2020 Page 157
Answer Keys Day 156:
1. 𝑓(𝑥) = (𝑥2 + 2𝑝𝑥) − 𝑟
2. (𝑥2 + 2𝑝𝑥) − 𝑟 = 0
𝑥2 + 2𝑝𝑥 = 𝑟
3. 𝑥2 + 2𝑝𝑥 + 𝑝2 = 𝑟 + 𝑝2
4. (𝑥 + 𝑝)2 = 𝑟 + 𝑝2
5. 𝑥 = −𝑝 ± √𝑟 + 𝑝2
Day 156 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 158
Use the following information to answer questions 1- 3
Consider the following equation: 𝑤2𝑥2 + 2𝑤𝑥 − 7 = 0
1. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
2. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
3. Solve the equation in number 2 for x.
Use the following information to answer questions 4 - 6
Consider the following equation: 𝑤𝑥2 − 2𝑤𝑥 + 4 = 0
4. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
5. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
6. Solve the equation in number 5 for x.
Day 156 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 159
Use the following information to answer questions 7 - 9
Consider the following equation: 𝑟𝑥2 − 2𝑟2𝑥 − 𝑐 = 0
7. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
8. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
9. Solve the equation in number 8 for x.
Use the following information to answer questions 7 - 9
Consider the following equation: 𝑘
4𝑥2 − 2𝑥 − 𝑐 = 0
10. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
11. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
12. Solve the equation in number 11 for x.
Day 156 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 160
Use the following information to answer questions 13 - 15
Consider the following equation: 25𝑘2𝑥2 − 2𝑘𝑥 + 𝑐 = 0
13. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
14. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
15. Solve the equation in number 14 for 𝑥.
Use the following information to answer questions 16 - 18
Consider the following equation: 25𝑥2 +𝑘
8𝑥 − 𝑘 = 0
16. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
17. Express it in the form (𝑥 + 𝑙)2 = 𝑚 where 𝑙 and 𝑚 are non-zero constants.
18. Solve the equation in number 17 for 𝑥.
Day 156 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 161
Use the following information to answer questions 19 - 20
Consider the following equation: 𝑝2𝑥2 + 2𝑝𝑥 = 0
19. Express it in the form 𝑎(𝑥2 + 𝑠𝑥 + 𝑡) = 𝑛 where 𝑎, 𝑠, 𝑡 and 𝑛 are non-zero constants.
20. Solve the equation in number 19 for 𝑥.
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HighSchoolMathTeachers©2020 Page 162
Answer Keys Day 156:
1. 𝑤2 (𝑥2 +2
𝑤𝑥 +
1
𝑤2) = 8
2. (𝑥 +1
𝑤)
2=
8
𝑤2
3. 𝑥 =−1±2√2
𝑤
4. 𝑤(𝑥2 − 2𝑥 + 1) = 𝑤 − 4
5.(𝑥 − 1)2 =𝑤−4
𝑤
6.𝑥 = 1 ± √4−𝑤
𝑤
7. 𝑟(𝑥2 − 2𝑟𝑥 + 𝑟2) = 𝑐 + 𝑟3
8. (𝑥 − 𝑟)2 =𝑐+𝑟3
𝑟
9. 𝑥 = 𝑟 ± √𝑐+𝑟3
𝑟
10. 𝑘
4(𝑥2 −
8
𝑘𝑥 + (
4
𝑘)
2
) = 𝑐 +4
𝑘
11. (𝑥 −4
𝑘)
2=
4(𝑘𝑐+4)
𝑘2
12.𝑥 =4
𝑘±
2√𝑘𝑐+4
𝑘
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HighSchoolMathTeachers©2020 Page 163
13. 25𝑘2 (𝑥2 −2
25𝑘𝑥 +
1
(25𝑘)2 ) =
1−25𝑐
25
14. (𝑥 −1
25𝑘 )
2=
1−25𝑐
625𝑘2
15. 𝑥 =1
25𝑘
±√1−25𝑐
25𝑘
16. 25 (𝑥2 +𝑘
200𝑥 + (
𝑘
400)
2 ) =
6400𝑘+𝑘2
6400
17. (𝑥 +𝑘
400)
2 =
6400𝑘+𝑘2
160000
18. 𝑥 = −𝑘
400±
√6400𝑘+𝑘2
400
19. 𝑝2 (𝑥2 +2
𝑝𝑥 +
1
𝑝2) = 1
20. 𝑥 = −1
𝑝±
1
𝑝
= 0, −2/𝑝
Day 156 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 164
Find the constant needed to complete the square for the expression 𝑎𝑥2 + 2𝑎𝑏𝑥
Day 156 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 165
Answer Keys Day 154:
𝑎𝑥2 + 2𝑎𝑏𝑥 + 𝑎2𝑏2
Day 157 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 166
1. Use the quadratic formula to solve the following equations, rounding to four significant figures where
necessary.
(i). 4𝑥2 − 6𝑥 − 1
(ii). (4𝑥)2 − 12𝑥 − 8
(ii). 𝑥2 − 6𝑥 + 9
(iii). 2𝑥2 + 13𝑥 + 15
Day 157 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 167
Answer Key
Day 157
1. (i). 𝑥 = 1.651, 𝑥 = −0.1514
(ii). 𝑥 = 1.175, 𝑥 = −0.4254
(ii). 𝑥 = 3
(iii). 𝑥 = −5, 𝑥 = −1.5
Day 157 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 168
A mason has 45 rectangular floor tiles, exactly enough to tile a floor. There should be 4 more
tiles along the length than along the width. The area of the room to be tiled is 270 square feet.
Find the dimensions of each tile if the length is 1.5 times the width.
1. Get a hard paper and cut it into 45 equal pieces such that the length of each piece is 1.5
times its width.
Arrange the paper pieces with all combinations to help you identify one that would give us 4
more pieces along the length that along the width.
2. What is the number of the paper pieces along the length and along the width?
3. What is the total number of the papers arranged? Identify the area of the floor.
Day 157 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 169
4. What is the area of each tile?
5. Using ratio of the length to the width, find the dimensions of each tile
Day 157 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 170
Answer Keys Day 157:
1.
2. Length - 9 tiles Width - 5 tiles
3. Total number of tiles = 45 Area to be tiled 270 sq. ft
4. The area of each tile = 6sq. ft 5. Each tile is 3 feet long and 2 feet wide.
Day 157 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 171
Use the following information to answer questions 1 - 5
A soccer ball sits at the edge of a kennel. Karl kicks the ball into the air and it follows a curve
given by the equation 𝒚(𝒙) = −𝟒
𝟑𝒙𝟐 + 𝟖𝒙 + 𝟐𝟗.
1. Determine the height of the kennel.
2. Find the horizontal distance moved when it reached when maximum height.
3. Find the maximum height reached.
4. Find the maximum horizontal distance moved.
5. Interpret the negative value of 𝑥 when 𝑦 = 0.
Day 157 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 172
Use the following information to answer questions 6 and 7
The product of two consecutive numbers is 1260.
6. Taking 𝑥 as the largest number, form an equation representing the situation.
7. Determine the two numbers.
Use the following information to answer questions 8 - 11
The product of the digits of a two digit number is 16. When the digits of the number are
interchanged the value of the new number is 54 less than the original number.
8. Write two equations representing the situation.
9. Write one equation that would lead to solving the equation.
10. What is the value of the original number?
Day 157 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 173
11. What is the difference between the new number (with the interchanged digits) and the
solution of 64 − 𝑥2 = 0.
12. The length of a rectangle is 5 units more than the width. If the area is 84 sq. ft. find the
width of the rectangle.
Use the following information to answer questions 13 - 16
A boy standing on a bridge throws a stone into the in the air after which it falls into the
stream below. The height of the stone in feet is 𝒔(𝒕) = −𝒕𝟐 +𝟏𝟏
𝟐𝒕 + 𝟏𝟓, where t is the time
taken in seconds.
13. Determine the height of the bridge.
14. Find the time it takes for the stone to reach the maximum height.
15. Determine the maximum height attained by the stone.
Day 157 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 174
16. Find the time it takes for the stone to reach the stream.
17. Find the perimeter of a square room whose area is 225 sq. ft.
Use the following information to answer questions 18 - 20
A fisherman on a cliff 60 ft high fires bullet to hit a shark in the sea. The vertical height of the
bullet in feet is 𝒔(𝒕) = 𝒌 − 𝒕𝟐 + 𝟐𝟖𝒕, where t is the time taken in seconds.
18. Determine the value of 𝑘.
19. Find the time it takes for the bullet to reach the maximum height.
20. How long did it take to hit the shark given that he was on target?
Day 157 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 175
Answer Keys Day 157:
1. 29
2. 3
3. 41
4. 8.55
5. It is the distance behind the initial position of the ball where Karl would need to be located if
the ball was being kicked from the ground instead of from the top of the kennel.
6. 𝑥2 − 𝑥 = 1260
7. 35 and 36 or -35 and -36
8. 10𝑥 + 1𝑦 − 54 = 1𝑥 + 10𝑦
9. 𝑥2 − 6𝑥 − 16 = 0
10. 82
11. 36 or 20
12. 7 ft
13. 15 ft
14. 11/4 sec
15. 22.56 ft
16. 15/2 sec
17. 60ft
18. 60
19. 14 sec
20. 30sec
Day 157 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 176
Determine the perimeter of a rectangular sign whose width is 6 in. less than the length given
that the area is 72 sq. in.
Day 157 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 177
Answer Keys Day 154:
1. 36 in
Day 158 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 178
1. Solve the following simultaneous equations
(i). 2𝑥 + 2𝑦 = 6; 6𝑥 − 𝑦 = 4
(ii). 5𝑥 + 3𝑦 = −1; 4𝑦 − 5𝑥 = −23
(iii). 𝑥 + 𝑦 = −1; 6𝑥 + 7𝑦 = −12
2. Find the coordinates of the points of intersection of the following curves:
(i) 𝑦 = 𝑥2 − 7𝑥 + 10, 𝑦 = 0
(ii). 𝑦 = 𝑥2 − 7𝑥 + 12, 𝑦 = 0
(iii).𝑦 = 6𝑥2 − 16𝑥3 − 1; 𝑥 = 0
Day 158 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 179
Answer Key
Day 158
1.(i) 𝑥 = 1, 𝑦 = 2
(ii). 𝑥 = −2, 𝑦 = 3
(iii).𝑥 = 5, 𝑦 = −6
2. (i). (5,0), (2,0)
(ii). (3,0), (4,0)
(iii). (0, −1)
Day 158 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 180
Consider the following function 𝑦 = 𝑥2 − 4𝑥 + 9 and 𝑦 + 1 = 3𝑥
1. Come up with a table of values for 𝑦 = 𝑥2 − 4𝑥 + 9 on −4 ≤ 𝑥 ≤ 3
𝑥
𝑦
2. Come up with a table of values for 𝑦 + 1 = 3𝑥 on −4 ≤ 𝑥 ≤ 3
𝑥
𝑦
3. Plot both functions on the graph grid below:
Day 158 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 181
4. How many points of intersections do the graphs have?
5. Identify the points of intersection
6. If the quadratic curve were to be vertically flipped across its vertex, would it intersect the line
graph? How many times?
Day 158 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 182
Answer Keys Day 158:
1.
𝑥 -4 -3 -2 -1 0 1 2 3
𝑦 41 30 21 14 9 6 5 6
2.
𝑥 -4 -3 -2 -1 0 1 2 3
𝑦 -13 -10 -7 -4 -1 2 5 8
3.
4. 2
5. (2,5) and (5,14)
6. Yes, they will intersect each other twice
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 183
Identify the number of solutions the systems have for questions 1-7.
1.
2.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 184
3.
4.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 185
5.
6.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 186
7.
Identify the solutions of the systems of equations shown in questions 8-14.
8.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 187
9.
10.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 188
11.
12.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 189
13.
14.
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 190
Use the following information to answer questions 15 – 20
Draw the graphical representation of the following systems and find their solution.
15. 𝑦 = 𝑥2 − 𝑥 − 3; 𝑦 = 𝑥 − 3
16. 𝑦 = 3𝑥 + 1; 𝑦 = 𝑥2 + 1
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 191
17. 𝑥 + 6 = 2𝑦; 2𝑦 = 𝑥2 − 4𝑥 + 10
18. 𝑦 =𝑥
2− 1; 𝑦 =
1
4(𝑥 − 2)2 − 2
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 192
19. 𝑦 = −𝑥 − 2; 𝑦 =1
2𝑥2 − 2𝑥 − 2
20. 𝑦 = −2𝑥; 𝑦 = 𝑥2 + 2𝑥
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 193
Answer Keys Day 158:
1. 2 2. 1 3. 1 4. 2 5. 1 6. 2 7. None 8. 𝑥 = −1, 𝑦 = −1
𝑥 = 0, 𝑦 = 0 9. 𝑥 = −2, 𝑦 = 0
𝑥 = 1, 𝑦 = 3 10. 𝑥 = 0, 𝑦 = 0 11. No solution 12. 𝑥 = −1, 𝑦 = −1
𝑥 = 3, 𝑦 = 3 13. 𝑥 = −2, 𝑦 = −2 14. 𝑥 = 0, 𝑦 = −2
𝑥 = 2, 𝑦 = 0 15. 𝑥 = 0, 𝑦 = −3
𝑥 = 2, 𝑦 = −1
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 194
16. 𝑥 = 0, 𝑦 = 1
𝑥 = 3, 𝑦 = 10
17. 𝑥 = 1, 𝑦 = 3.5 𝑥 = 4, 𝑦 = 5
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 195
18. 𝑥 = 0, 𝑦 = −1 𝑥 = 6, 𝑦 = 2
19. 𝑥 = 0, 𝑦 = −2 𝑥 = 2, 𝑦 = −4
Day 158 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 196
20. 𝑥 = −4, 𝑦 = 8 𝑥 = 0, 𝑦 = 0
Day 158 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 197
Find the solution of the following system of equations
𝑦 = 𝑥2 − 4𝑥 + 4
𝑦 − 𝑥 + 2 = 0
Day 158 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 198
Answer Keys Day 158:
𝑥 = 2, 𝑦 = 0
𝑥 = 3, 𝑦 = 1
Day 159 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 199
Solve the following system of equations
1. 𝑥 + 𝑦 = 12
𝑥 − 𝑦 = −2
2. 2𝑥 + 3𝑦 = 19
4𝑥 − 5𝑦 = 5
3. 𝑡
𝑟= −2
9𝑟 − 2𝑡 = 39
Day 159 Bellringer Name ____________________________________
HighSchoolMathTeachers©2020 Page 200
Answer Key
Day 159
1. 𝑥 = 5, 𝑦 = 7
2. 𝑥 = 5, 𝑦 = 3
3. 𝑡 = −6, 𝑟 = 3
Day 159 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 201
A man fires a bullet into the air and it follows a path described by 𝑠 − 70 + 𝑥2 = 42𝑥. If the
target is along the line 𝑠 + 10 = 4𝑥, determine the position of the target if the distance units is
feet.
1. Solve the linear function for s.
2. Substitute for 𝑠 in the quadratic formula using the equation above.
3. Form a standard quadratic equation from the resultant equation.
4. Solve the quadratic equation in 3 above.
5. Identify the horizontal distance the target is from the point where the bullet is fired.
6. Using the equation In 1 above, find the height of the target.
Day 159 Activity Name ____________________________________
HighSchoolMathTeachers©2020 Page 202
Answer Keys Day 159:
1. 𝑠 = 4𝑥 − 10
2. 4𝑥 − 10 − 70 + 𝑥2 = 42𝑥
3. 𝑥2 − 38𝑥 − 80 = 0
4. 𝑥 = −2, 𝑥 = 40
5. 40 𝑓𝑒𝑒𝑡
6. 150 𝑓𝑒𝑒𝑡
Day 159 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 203
Identify the number of solutions for the systems in questions 1-5.
1. 𝑦 = 𝑥 + 4; 𝑦 = 𝑥2 − 2𝑥 + 6
2.𝑦 = 𝑥 + 3; 𝑦 = 3𝑥2 + 2𝑥 + 4
3. 𝑦 = 6𝑥 + 4; 𝑦 = 3𝑥2 + 6𝑥 + 4
4. 𝑦 = −2𝑥2 + 4𝑥 + 2; 𝑦 = 4
5. 𝑥 = 2, 𝑦 = −20𝑥2 − 4𝑥 + 4
Find the solution of the systems shown in 6 –20
6. 𝑦 = 𝑥, 𝑦 = −𝑥2 + 𝑥 + 4
Day 159 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 204
7. 𝑦 = 𝑥 + 3, 𝑦 + 𝑥2 = 𝑥 + 4
8. 𝑦 = 2𝑥 + 3, 𝑦 = −2𝑥2 + 𝑥 + 4
9. 𝑦 = 2𝑥 − 4, 𝑦 = −2𝑥2 + 4𝑥 + 8
10. 𝑦 = 𝑥 − 4; 𝑦 = 𝑥2 + 4𝑥 − 8
11.𝑦 = 2𝑥 + 8, 𝑦 = 𝑥2 + 4𝑥
12. 𝑦 = 2𝑥 − 2, 𝑦 = 𝑥2 + 𝑥 − 2
Day 159 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 205
13. 𝑦 = −3𝑥 − 3, 𝑦 = 3𝑥2 − 6𝑥 − 3
14. 𝑦 = −1
2 𝑥 − 3; 𝑦 =
1
2𝑥2 + 𝑥 − 2
15.𝑦 = −1, 𝑦 = −𝑥2 + 2𝑥 + 2
16. 𝑦 =1
2𝑥2 + 3𝑥 − 2, 𝑦 = 6
17. 𝑦 = 2𝑥 + 6, 𝑦 = 4𝑥2 + 16𝑥 − 2
18. 𝑦 = 2𝑥 + 2, 𝑦 = 𝑥2 + 2𝑥 − 2
Day 159 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 206
19. 2𝑥 + 8, 𝑦 = 𝑥2 + 𝑥 − 4
20. 𝑦 = 𝑥 − 5, 𝑦 = −𝑥2 + 𝑥 + 4
Day 159 Practice Name ____________________________________
HighSchoolMathTeachers©2020 Page 207
Answer KeysDay 159:
1. 2 2. None 3. 1 4. 1 5. 1 6. 𝑥 = 2, 𝑦 = 2
𝑥 = −2, 𝑦 = −2 7. 𝑥 = 1, 𝑦 = 4
𝑥 = −1, 𝑦 = 2 8. 𝑥 = 0.5, 𝑦 = 4
𝑥 = −1, 𝑦 = 1 9. 𝑥 = 3, 𝑦 = 2
𝑥 = −2, 𝑦 = −8 10. 𝑥 = 1, 𝑦 = −3
𝑥 = −4, 𝑦 = −8 11. 𝑥 = −4, 𝑦 = 0
𝑥 = 2, 𝑦 = 12 12. 𝑥 = 1, 𝑦 = 0
𝑥 = 0, 𝑦 = −2 13. 𝑥 = 0, 𝑦 = −3
𝑥 = 1, 𝑦 = −6 14. 𝑥 = −1, 𝑦 = −2.5
𝑥 = −2, 𝑦 = −2 15. 𝑥 = −1, 𝑦 = −1
𝑥 = 3, 𝑦 = −1 16. 𝑥 = −8, 𝑦 = 6
𝑥 = 2, 𝑦 = 6 17. 𝑥 = 0.5, 𝑦 = 7
𝑥 = −4, 𝑦 = −2 18. 𝑥 = −2, 𝑦 = −2
𝑥 = 2, 𝑦 = 6 19. 𝑥 = 4, 𝑦 = 16
𝑥 = −3, 𝑦 = 2 20. 𝑥 = 3, 𝑦 = −2
𝑥 = −3, 𝑦 = −8
Day 159 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 208
Determine the value of 𝑥 and 𝑦 that satisfy both equations
𝑦 = 4𝑥 − 6
𝑦 = 𝑥2 − 4𝑥 + 6
Day 159 Exit Slip Name ____________________________________
HighSchoolMathTeachers©2020 Page 209
Answer Keys Day 159:
𝑥 = 𝑦 = 2
𝑥 = 6, 𝑦 = 18
210
High School Math Teachers
Algebra 1
Weekly Assessment Package
Week 32
HighSchoolMathTeaches©2020
211
Week 32
Weekly Assessments
212
Week #32 - Systems and Quadratic Formula
1. Use the quadratic formula to determine
the solutions for the following quadratic
functions if possible. Round your
answers to the nearest hundredth if
necessary.
a. 𝑥2 − 9𝑥 − 8 = 0
b. 10𝑥2 + 6𝑥 + 2 = 0
c. −3𝑥 + 2 = −2𝑥2 + 5𝑥 − 6
d. −0.25𝑥2 + 1.62𝑥 − 3.39 = 0
2. Solve each quadratic using any
method. Explain your choice of
method for solving.
a. −𝑥2 − 14 = 𝑥2 + 10𝑥
b. 𝑥2 = −3𝑥 + 1
c. (𝑥 + 9)2 − 64 = 0
d. (2𝑥 + 5)(−4𝑥 + 7) = 0
3. Solve the following systems using any method.
a. {𝑓(𝑥) = −2𝑥 + 4
𝑔(𝑥) = −2𝑥2 + 3𝑥 − 2
b. {𝑓(𝑥) =
1
3𝑥 − 2
𝑔(𝑥) = 𝑥2 − 3𝑥 − 2
c. {𝑓(𝑥) = −6
𝑔(𝑥) = 4𝑥2 + 4𝑥 − 5
d. {𝑓(𝑥) = −
2
5𝑥 + 3
𝑔(𝑥) = −2𝑥2 − 4𝑥 + 9
213
Week 32 - KEYS
Weekly Assessments
214
Week #32 - Systems and Quadratic Formula Answer Key 1. Use the quadratic formula to determine
the solutions for the following quadratic
functions. Round your answers to the
nearest hundredth if necessary.
a. 𝑥2 − 9𝑥 − 8 = 0
𝒙 = −𝟎. 𝟖𝟐, 𝟗. 𝟖𝟐 b. 10𝑥2 + 6𝑥 + 2 = 0
No real solutions c. −3𝑥 + 2 = −2𝑥2 + 5𝑥 − 6
𝒙 = 𝟐 d. −0.25𝑥2 + 1.62𝑥 − 3.39 = 0
No real solutions
2. Solve each quadratic using any
method. Explain your choice of
method for solving.
Explanations of method chosen will vary a. −𝑥2 − 14 = 𝑥2 + 10𝑥
No real solution
b. 𝑥2 = −3𝑥 + 1
𝒙 = −𝟑. 𝟑𝟎, 𝟎. 𝟑𝟎
c. (𝑥 + 9)2 − 64 = 0
𝒙 = −𝟏. 𝟏𝟕
d. (2𝑥 + 5)(−4𝑥 + 7) = 0
(-2.5, 0) and (1.75, 0) 3. Solve the following systems using any method. Round answers to the nearest
hundredth if necessary.
a. {𝑓(𝑥) = −2𝑥 + 4
𝑔(𝑥) = −2𝑥2 + 3𝑥 − 2
𝑵𝑶 𝑺𝑶𝑳𝑼𝑻𝑰𝑶𝑵 − 𝑮𝒓𝒂𝒑𝒉𝒔 𝒅𝒐𝒏′𝒕 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕
b. {𝑓(𝑥) =
1
3𝑥 − 2
𝑔(𝑥) = 𝑥2 − 3𝑥 − 2
(𝟎, −𝟐) 𝒂𝒏𝒅 (𝟑. 𝟑𝟑, −𝟎. 𝟖𝟗)
c. {𝑓(𝑥) = −6
𝑔(𝑥) = 4𝑥2 + 4𝑥 − 5
(−𝟎. 𝟓, −𝟔)
d. {𝑓(𝑥) = −
2
5𝑥 + 3
𝑔(𝑥) = −2𝑥2 − 4𝑥 + 9
(−𝟐. 𝟖𝟓, 𝟒. 𝟏𝟒) 𝒂𝒏𝒅 (𝟏. 𝟎𝟓, 𝟐. 𝟓𝟖)
Unit 12 Test Name ____________________________________
HighSchoolMathTeachers ©2020 Page 215
Questions:
1. Find the square of the following:
a) 2
b) -7
c) -13
2. Determine the square root of the following:
a) 16
b) 64
c) 144
3. Solve the equation:
𝑥2 = 576
4. Solve the equation:
(𝑥 − 5)2 = 36
5. How many solutions does each quadratic equation have?
6. Expand the following:
a) 2𝑥(3 + 5𝑥)
b) (1 + 3𝑥)(𝑥 − 2)
7. Factor out the following:
a) 4𝑥 − 12
b) 18𝑥 − 27𝑥2
Unit 12 Test Name ____________________________________
HighSchoolMathTeachers ©2020 Page 216
8. Factor the following:
a) 𝑡2 + 4𝑥 − 5
b) 2𝑏2 + 5𝑏 − 12
9. List all 8 steps of using the Box method of factoring a polynomial.
10. Find the x-intercepts of the function:
𝑓(𝑥) = 𝑥2 − 2𝑥 − 24
11. Write the expression 𝑥2 − 2𝑥 − 24 in the form (𝑥 − ℎ)2 + 𝑘 by completing the square.
12. Solve the equation:
9𝑡2 − 54 = −45𝑡
13. Graph the quadratic function:
𝑓(𝑥) = 2𝑥2 − 5𝑥 − 3
Unit 12 Test Name ____________________________________
HighSchoolMathTeachers ©2020 Page 217
14. Express the following expression in the form 𝑦 = 𝑎(𝑥 − ℎ) + 𝑘:
𝑦 = −9𝑥 + 7 + 3𝑥2
15. If 𝑓(𝑥) = 2𝑥2 − 4𝑥 + 12, draw a table of values for 𝑥 and 𝑓(𝑥)for integer values in the range
−2 ≤ 𝑥 ≤ 2.
𝑥
𝑓(𝑥)
16. Make 𝑥 the subject of the formula:
(𝑥 − 𝑟)2 =16
25𝑡4
17. Write the quadratic formula!
18. Use the quadratic formula to determine the solutions for the following quadratic function if
possible:
5𝑥2 + 3𝑥 + 1 = 0
19. The product of two consecutive numbers is 1806. Determine the numbers!
20. The graph of a quadratic relation is:
a) A straight line
b) A circle
c) A parabola
Unit 12 Test Name ____________________________________
HighSchoolMathTeachers ©2020 Page 218
Answers:
1. a) 4
b) 49
c) 169
2. a) √16 = ±4
b) √64 = ±8
c) √144 = ±12
3. 𝑥 = ±24
4. 𝑥1 = 11; 𝑥2 = −1
5. Each quadratic equation has two solutions.
6.
a) 6𝑥 + 10𝑥2
b) 3𝑥2 − 5𝑥 − 2
7.
a) 4(𝑥 − 3)
b) 9𝑥(2 − 3𝑥)
8.
a) (𝑡 − 1)(𝑡 + 5)
b) (2𝑏 − 3)(𝑏 + 4)
9.
Step 1 - Place the first and last term in the box
Step 2 - List factors
Step 3 - Choose factors
Step 4 - Place the factors in the box
Step 5 - Find the greatest common factor
Step 6 - Use multiplication
Step 7 - Fill in remain boxes
Step 8 - Write the factors
10. 𝑥1 = −4 and 𝑥2 = 6
11. (𝑥 − 1)2 − 25
12. 𝑡1 = −6 and 𝑡2 = 1
Unit 12 Test Name ____________________________________
HighSchoolMathTeachers ©2020 Page 219
13.
14. 𝑦 = 3(𝑥 −3
2)2 +
1
4
15.
𝑥 -2 -1 0 1 2
𝑓(𝑥) 28 18 12 10 12
16. 𝑥 =4
5𝑡2 + 𝑟
17.
18. This function doesn’t have real solutions.
19. 42 and 43
20. c) A parabola