Complete Week 31 Package - High School Math Teachers · The numbers of x’s down one side of the...

Complete Week 31

Package

Jeanette Stein

Table of Contents

Unit 12 Pacing Chart -------------------------------------------------------------------------------------------- 1

Day 151 Bellringer -------------------------------------------------------------------------------------------- 3

Day 151 Activity -------------------------------------------------------------------------------------------- 5

Day 151 Practice -------------------------------------------------------------------------------------------- 16

Day 151 Exit Slip -------------------------------------------------------------------------------------------- 29

Day 152 Bellringer -------------------------------------------------------------------------------------------- 31

Day 152 Activity -------------------------------------------------------------------------------------------- 33

Day 152 Practice -------------------------------------------------------------------------------------------- 36

Day 152 Exit Slip -------------------------------------------------------------------------------------------- 38

Day 153 Bellringer -------------------------------------------------------------------------------------------- 40

Day 153 Activity -------------------------------------------------------------------------------------------- 42

Day 153 Practice -------------------------------------------------------------------------------------------- 44

Day 153 Exit Slip -------------------------------------------------------------------------------------------- 49

Day 154 Bellringer -------------------------------------------------------------------------------------------- 51

Day 154 Activity -------------------------------------------------------------------------------------------- 53

Day 154 Practice -------------------------------------------------------------------------------------------- 56

Day 154 Exit Slip -------------------------------------------------------------------------------------------- 62

Weekly Assessment -------------------------------------------------------------------------------------------- 64

CCSS Algebra 1 Pacing Chart – Unit 12

Algebra1Teachers @ 2015 Page 1

Unit Week Day CCSS Standards Mathematical Practices Objective I Can Statements

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.



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.



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).

CCSS Algebra 1 Pacing Chart – Unit 12




155 Assessment Assessment Assessment Assessment

Day 151 Bellringer Name ____________________________________


Factor each of the following perfect square trinomials. In the last two problems, look for a greatest

common factor to remove first.

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)

Day 151 Activity Name ____________________________________


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 quadratic 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 what we need or keeping whatever extra pieces we

may get. 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 represent𝑥2 + 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) Fill in the blanks 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) Fill in the blanks with numbers to make the statement true: 𝑥2 − 6𝑥 + 𝑐 = (𝑥+? )2



6. What is the relationship between the coefficient of x 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 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 x 2 -tile in each square.

• You must use all the x 2 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

Steps: Example: 𝑥2 + 6𝑥 − 14 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0

a) Move constant term of quadratic to the other side. Write the equation 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______ 𝑓𝑜𝑟𝑚

𝑥2 + 6𝑥 − 14 + 14 = 0 + 14 𝑥2 + 6𝑥 + _____ = 14 + _____

b) Complete the square by adding a constant to both sides.

𝑥2 + 6𝑥 + 9 = 141 + 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 𝑥2term

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=2 W=2 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=3 W=3 c) C=4 ?=-2

5.

a) We need 9 unit tiles to complete the square. b) L=4 W=4 c) C=9 ?=-3

6. The number of x’s you have done 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 the 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 − 14 = 0 You try: 𝑥2 + 8𝑥 − 20 = 0

a) Move constant term of quadratic to the other side. Write the equation 𝑎𝑥2 + 𝑏𝑥 + ______ = −𝑐______ 𝑓𝑜𝑟𝑚

𝑥2 + 6𝑥 − 14 + 14 = 0 + 14 𝑥2 + 6𝑥 + _____ = 14 + _____

𝑥2 + 8𝑥 − 20 + 20 = 0 + 20

b) Complete the square by adding a constant to both sides.

𝑥2 + 6𝑥 + 9 = 141 + 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 hence use it to write an expression representing

them if red tiles implies negative and green 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 − 6𝑥 + 1

10. 3𝑥2 + 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

or 16𝑥2 + 16𝑥 + 4



15. 4

16. 6

17. 6

18. 4

19. 9

20. 9

Day 151 Exit Slip Name ____________________________________


Draw color tiles representing the quadratic expression 𝑥2 − 4𝑥 hence complete it




𝑥2 − 4𝑥 + 4



Express the following expressions in the form 𝑦 = 𝑎(𝑥 − ℎ) + 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 expressions𝑓(𝑥) = 8𝑥 − 2𝑥2 − 3

1. Draw a table of value 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 from which the value in 2 occurs.

4. Express the expression above in the standard form of a quadratic function.

5. Enclose the first two terms in bracket and factorize appropriately so that the coefficient of 𝑥2

is 1.

6. Complete the square in the bracket



7. Express the resultant in 6 above so that you have an expression of the form 𝑎(𝑥 − ℎ) + 𝑘

where 𝑎 can be positive or negative.

8. Compare the value of 𝑘 and ℎ in 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 9. When a quadratic expression is expressed in the form 𝑎(𝑥 − ℎ) + 𝑘 where 𝑎 < 0, the value of

𝑘 is the maximum valueof 𝑓(𝑥) whilethe value of ℎ is the𝑥 −coordinate of the point where the maximum value occurs.

Day 152 Practice Name ____________________________________


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 expression

𝑦 = 1 − 3𝑥2 + 18𝑥




−3(𝑥 − 3)2 + 28

The minimum value is 28



1. Make 𝑥 the subject of the formula in the following

(i). 2𝑥 + 𝑡 = 𝑝𝑥

(ii). 𝑥

5+ 𝑠 =

4

2𝑥

(iii). 𝑡𝑥

𝑚+ 𝑟𝑥 = 1

2. Complete the square then factorize

(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. Factorize 4 from the left hand side

3. Complete the square in brackets in 2 above

4. Simply the bracket so that you have an equation of the form 𝑎(𝑥2 + 𝑑𝑥 + 𝑒2) = 𝑓.

5. Factorize the square in the bracket using appropriate method

6. Make 𝑥 the subject of the formula.




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



Make 𝑥 the subject of the formula, simplify completely 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

𝑝

17. 𝑥 =5

9𝑡 ± 1.732

18. 𝑥 =3

7𝑟 ± √

4𝑟+3

2

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. Use 𝑥 the subject of the formula in the following

(i). (𝑥 − 4)2 = 9𝑡

(ii). (2𝑥 + 𝑟)2 = 16

2. Find the coordinates of the vertex of the following 2𝑥2 − 24𝑥 + 9



Answer Key

Day 154

1. (i) 𝑥 = 4 ± 3√𝑡

(ii). 𝑥 = −𝑟

2± 2

2. (−6, −63)



Consider the following function 8𝑥2 + 4𝑥 − 1

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. Complete the square if the expression in the first bracket

3. Factorize the expression in the first bracket (completed square)

4. Express the expression in 3 above in terms of 𝑟(𝑥 − ℎ) + 𝑘 where 𝑟 ≠ 0, ℎ and 𝑘 are

constants.



5. Identify the relative maximum or the minimum value of the expression.

6. Identify the vertex of the function

7. Equate the function to zero then determine the value of 𝑥.




1. (8𝑥2 + 4𝑥) − 1

2. 8 (𝑥2 +1

2𝑥 + (

1

4)

2

) −1

2− 1

3. 8 (𝑥 +1

2)

2

−1

2− 1

4. 8 (𝑥 +1

2)

2

−3

2

5. Relative minimum is −3

2

6. Vertex(−1

2, −

3

2 )

7. 8 (𝑥 +1

2)

2

−3

2= 0

𝑥 = −1

2±

√3

4



Make 𝑥 the subject of the formula, simplify completely where possible from 1 - 5

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



Complete the following squares 6 – 13

6. 𝑥2 − 8𝑥

7. 𝑥2 − 22𝑥

8. 4𝑥2 − 40𝑥

9. 81𝑥2 − 36𝑡2𝑥

10. 4𝑥2 + 8𝑥



11. 𝑝2𝑥2 + 2𝑝𝑥

12. 𝑎2𝑥2 − (4𝑝)2𝑥

13. 25𝑥2 +𝑎

2𝑠𝑥

14. 36𝑥2 − 4𝑟𝑥



For questions 15 – 17, draw square tiles representing the 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 expressions.

17. 𝑥2 + 4𝑥 + 2

18. 2𝑥2 − 9𝑥 + 3

19. −4𝑥2 + 8𝑥 + 3

20.−3𝑥2 + 7𝑥 + 8




1. 𝑥 = 2𝑑 ± 3

2. 𝑥 = 6𝑝 ± 5𝑝

3. 𝑥 = −2 ±5

12

4. 𝑥 = −3

2𝑠 ±

1

2𝑛

5. 𝑥 =𝑟±𝑞

2

6. 𝑥2 − 8𝑥 + 16

7. 𝑥2 − 22𝑥 + 121

8. 4𝑥2 − 40𝑥 + 400

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 −65

6

19. Maximum value 7

20. Maximum value 49

12



Express find the minimum or the maximum of 𝑦 = 2𝑥2 + 14𝑥 − 2




𝑦 = 2 (𝑥 +7

2)

2

−53

2

The function has a minimum at −53

2.

Algebra 1 Teachers

Weekly Assessment Package

Unit 12

Created by:

Jeanette Stein

& Phill Adams

©2015 Algebra 1 Teachers

65 Semester 2 Skills | Algebra1Teachers.com

Algebra 1 Common Core

Semester 2 Skills

Week Unit CCSS Skill

31 12 A.SSE.3b Complete the square

31 12 F.IF.8 Use completing the square to find maximum and minumum values

66 | Algebra1Teachers.com

Unit 12 - Solve Quadratic Functions

Weekly Assessments

67 Unit 12 - Solve Quadratic Functions | Algebra1Teachers.com

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

68 Unit 12 - KEYS | Algebra1Teachers.com

Unit 12 - KEYS

Weekly Assessments

69 Unit 12 - KEYS | Algebra1Teachers.com

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)

b. 𝑓(𝑥) = 𝑥2 − 6𝑥 + 5

𝒇(𝒙) = (𝒙 − 𝟑)𝟐 − 𝟒 (𝟑, −𝟒)

c. 𝑓(𝑥) = 𝑥2 + 4𝑥

𝒇(𝒙) = (𝒙 + 𝟐)𝟐 − 𝟒 (−𝟐, −𝟒)

d. 𝑓(𝑥) = 6𝑥2 + 12𝑥 + 13

𝒇(𝒙) = 𝟔(𝒙 + 𝟏)𝟐 + 𝟕 (−𝟏, 𝟕)

Complete Week 31 Package - High School Math Teachers · The numbers of x’s down one side of the...

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