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Complete Unit 12

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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.


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.


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.


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.


30 – Factoring

150 Assessment Assessment Assessment Assessment




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.


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.



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.


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.



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).





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).





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.



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.





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).



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.




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 ____________________________________


Unit 12 Lesson Plans Name ____________________________________


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:




Course: Algebra 1


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:



Reflection:





Course: Algebra 1


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.






Reflection:





Course: Algebra 1


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.




Reflection:





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.


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




Technology:

Reflection:





Course: Algebra 1


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.


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




Technology:

Reflection:





Course: Algebra 1


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).






Technology:

Reflection:





Course: Algebra 1


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




Technology: Superscript: ⁻ ⁺ ⁼ ⁽ ⁾ ⁰ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ⁸ ⁹ Subscript: ₊ ₋ ₌ ₍ ₎ ₀ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉

Reflection:





Course: Algebra 1


Day: 155

Common Core State Standard: Assessment

Mathematical Practice: Assessment

Objective: Assessment

I can statement: Assessment

Procedures: Assessment

Materials: Weekly Assessment Unit 12


Technology:

Reflection:





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




Reflection:





Course: Algebra 1


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



5. Use the Day-157-Practice -Quadratic application practice as individual practice or homework.


Reflection:





Course: Algebra 1


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



5. Use the Day 158 –Practice- Systems of linear and quadratic equations (Day 1) as individual practice or homework.


Reflection:





Course: Algebra 1


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



5. Use the Day 152 –Practice- Systems of linear and quadratic equations (Day 2) as individual practice or homework.


Reflection:


Day 146 Bellringer Name ____________________________________


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



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 ____________________________________


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



Answer Key

1. a. 3 seconds

b. √11.25 𝑠𝑒𝑐𝑜𝑛𝑑𝑠;

c. √12.5 𝑠𝑒𝑐𝑜𝑛𝑑𝑠;

d. √5.9375 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Day 146 Practice Name ____________________________________


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



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



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

Day 146 Exit Slip Name ____________________________________


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



Answer Key

1. 13

2. 8.06

3. ±20

4. ±8

11

5. −11, 5



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



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)



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.



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.



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



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.



x

x

6

6


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



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)



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



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)



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



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)



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



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



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



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).



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.



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.



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.



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



Activity Worksheet

EXPLORE 1

3𝑥2 + 16 + 5

Product:___________Factors of the product:

3𝑥2 + 16𝑥 + 5=(___________________)(_________________)



EXPLORE 2

4𝑥2 + 5𝑥 − 6

Product:_____________Factors of the product:

4𝑥2 + 5𝑥 − 6 =

(________________)(_____________)



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



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)



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



𝑥 2

4𝑥

−3

4𝑥2 8𝑥

−3𝑥 −6

STEP 5

STEP 6

𝟒𝒙𝟐 + 𝟓𝒙 − 𝟔 = (𝟒𝒙 + (−𝟑))(𝒙 + 𝟐)

= (𝟒𝒙 − 𝟑)(𝒙 + 𝟐)



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)



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.



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.




1. 𝟐𝒙𝟐 + 𝟓𝒙 + 𝟐

2. 𝟐𝒂𝟐 + 𝟕𝒂 + 𝟔

3. 𝟒𝒙𝟐 − 𝟒𝒙 − 𝟏𝟓

4. 𝟗 + 𝟗𝒙 + 𝟐𝒙𝟐

5. 𝟖 + 𝟏𝟒𝒓 + 𝟑𝒓𝟐

6. 𝟏𝟐 − 𝟏𝟑𝒂 − 𝟒𝒂𝟐



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)



Factor each completely.

1) 3𝑝2 − 2𝑝 − 5

2) 3𝑛2 − 8𝑛 + 4



Answer Key

1) (𝟑𝒑 − 𝟓)(𝒑 + 𝟏) 2) (𝟑𝒏 − 𝟐)(𝒏 − 𝟐)



Factor.

1) 15𝑛2 − 27𝑛 − 6

2) 5𝑥2 − 18𝑥 + 9

3) 4𝑛2 − 15𝑛 − 25

4) 4𝑥2 − 35𝑥 + 49



Answer Key

1. 3(5𝑛 + 1)(𝑛 − 2)

2. (5𝑥 − 3)(𝑥 − 3)

3. (𝑛 − 5)(4𝑛 + 5)

4. (𝑥 − 7)(4𝑥 − 7)



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.



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?



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.



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



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



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



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}



Solve the quadratics.

1. 7𝑣2 − 42 = −35𝑣

2. 𝑘2 = −4𝑘 − 4



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



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



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)



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.



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



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



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



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



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)



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



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



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



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



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

Day 151 Practice Name ___________________________________


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



4. 𝑥2 − 10𝑥

5. 4𝑥2 + 4𝑥

6. 4𝑥2 − 4𝑥



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.



9.

10.

11.



12.

13.



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.



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.



19.

20.



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



9. 9𝑥2 − 12𝑥 + 4

10. 9𝑥2 + 6𝑥 + 1

11. 𝑥2 + 4𝑥 + 4

Or 𝑥2 − 4𝑥 + 4



12. 𝑥2 + 8𝑥 + 16

Or 𝑥2 − 8𝑥 + 16

13. 𝑥2 − 2𝑥 + 1

Or 𝑥2 + 2𝑥 + 1



14. 16𝑥2 − 16𝑥 + 4

15. 4

16. 6

17. 6

18. 4

19. 9

20. 9



Draw color tiles representing the quadratic expression 𝑥2 + 4𝑥 and show how you would

complete the square.




𝑥2 + 4𝑥 + 4



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



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



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



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.




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.



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



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}



Determine the minimum or the maximum value of the following equation.

𝑦 = 1 − 3𝑥2 + 18𝑥




𝑦 = −3(𝑥 − 3)2 + 28

The maximum value is 28



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



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



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.




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



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



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



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



19. 4(𝑥 + 3𝑟)2 = 19

20. 𝑟2 (𝑥 −3

4𝑡)

2

= 𝑡2




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



Express 𝑥 in terms of 𝑙 and 𝑚 if (𝑥 − 𝑙)2 = 𝑚




𝑥 = 𝑙 ± √𝑚



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



Answer Key

Day 154

1. (i) 𝑥 = 4 ± 3√𝑡

(ii). 𝑥 = −𝑟

2± 2

2. (6, −63)



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.



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.




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



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



For questions 6 – 14, compare the square:

6. 𝑥2 − 8𝑥

7. 𝑥2 − 22𝑥

8. 4𝑥2 − 40𝑥

9. 81𝑥2 − 36𝑥

10. 4𝑥2 + 8𝑥



11. 𝑝2𝑥2 + 2𝑝𝑥

12. 𝑎2𝑥2 − (4𝑝)2𝑥

13. 25𝑥2 +𝑎

2𝑠𝑥

14. 36𝑥2 − 4𝑟𝑥



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



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




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



Find the minimum or the maximum value of 𝑦 = 2𝑥2 + 14𝑥 − 2




𝑦 = 2 (𝑥 +7

2)

2

−53

2

The function has a minimum value of −53

2.


Algebra 1


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 ________________________________


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 ________________________________


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



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.




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



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







Consider the following equation: 𝑟𝑥2 − 2𝑟2𝑥 − 𝑐 = 0





Consider the following equation: 𝑘

4𝑥2 − 2𝑥 − 𝑐 = 0







Consider the following equation: 25𝑘2𝑥2 − 2𝑘𝑥 + 𝑐 = 0



15. Solve the equation in number 14 for 𝑥.


Consider the following equation: 25𝑥2 +𝑘

8𝑥 − 𝑘 = 0







Consider the following equation: 𝑝2𝑥2 + 2𝑝𝑥 = 0






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



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



Find the constant needed to complete the square for the expression 𝑎𝑥2 + 2𝑎𝑏𝑥




𝑎𝑥2 + 2𝑎𝑏𝑥 + 𝑎2𝑏2



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



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



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.



4. What is the area of each tile?

5. Using ratio of the length to the width, find the dimensions of each tile




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.




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.



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.


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?



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.


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.



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.


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?




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



Determine the perimeter of a rectangular sign whose width is 6 in. less than the length given

that the area is 72 sq. in.




1. 36 in



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



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)



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:



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?




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



Identify the number of solutions the systems have for questions 1-7.

1.

2.



3.

4.



5.

6.



7.

Identify the solutions of the systems of equations shown in questions 8-14.

8.



9.

10.



11.

12.



13.

14.



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



17. 𝑥 + 6 = 2𝑦; 2𝑦 = 𝑥2 − 4𝑥 + 10

18. 𝑦 =𝑥

2− 1; 𝑦 =

1

4(𝑥 − 2)2 − 2



19. 𝑦 = −𝑥 − 2; 𝑦 =1

2𝑥2 − 2𝑥 − 2

20. 𝑦 = −2𝑥; 𝑦 = 𝑥2 + 2𝑥




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



16. 𝑥 = 0, 𝑦 = 1

𝑥 = 3, 𝑦 = 10

17. 𝑥 = 1, 𝑦 = 3.5 𝑥 = 4, 𝑦 = 5



18. 𝑥 = 0, 𝑦 = −1 𝑥 = 6, 𝑦 = 2

19. 𝑥 = 0, 𝑦 = −2 𝑥 = 2, 𝑦 = −4



20. 𝑥 = −4, 𝑦 = 8 𝑥 = 0, 𝑦 = 0



Find the solution of the following system of equations

𝑦 = 𝑥2 − 4𝑥 + 4

𝑦 − 𝑥 + 2 = 0




𝑥 = 2, 𝑦 = 0

𝑥 = 3, 𝑦 = 1



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



Answer Key

Day 159

1. 𝑥 = 5, 𝑦 = 7

2. 𝑥 = 5, 𝑦 = 3

3. 𝑡 = −6, 𝑟 = 3



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.




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



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



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



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



19. 2𝑥 + 8, 𝑦 = 𝑥2 + 𝑥 − 4

20. 𝑦 = 𝑥 − 5, 𝑦 = −𝑥2 + 𝑥 + 4



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



Determine the value of 𝑥 and 𝑦 that satisfy both equations

𝑦 = 4𝑥 − 6

𝑦 = 𝑥2 − 4𝑥 + 6




𝑥 = 𝑦 = 2

𝑥 = 6, 𝑦 = 18

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


(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 ____________________________________


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.


9𝑡2 − 54 = −45𝑡

13. Graph the quadratic function:

𝑓(𝑥) = 2𝑥2 − 5𝑥 − 3

Unit 12 Test Name ____________________________________


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 ____________________________________


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 ____________________________________


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

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