1....Unit 3 – 2 Name: Solve the following QUADRATIC EQUATIONS using the SQUARE ROOT METHOD: 1. w2...

Name:

Factor the GCF from each expression

1. 54 315 xx 2. 2416 2 x

3. 556374 423618 yxyxyx 4. 3233 xxx

Not all may be possible.

1. Find two numbers that sum to 8 and have a product of 12 ____________________

2. Find two numbers that sum to 5 and have a product of 6 _____________________

3. Find two numbers that sum to 5 and have a product of -14 ___________________

4. Find two numbers that sum to -6 and have a product of 12 ___________________

5. Find two numbers that sum to 16 and have a product of 15 ___________________

6. Find two numbers that sum to -4 and have a product of -21 __________________


8. Find two numbers that sum to -14 and have a product of 40 __________________


10. Find two numbers that sum to 8 and have a product of 16 ___________________

11. Multiply the following:

a. x 6 x 3 b. x 7 x 2

x2 + _______ x + ________

Notice: What is the sum

of the constants in each

binomial above?

Notice: What is the

product of the constants

in each binomial above?

x2 + _______ x + ________

Notice: What is the sum

of the constants in each

binomial above?

Notice: What is the

product of the constants

in each binomial above?

M. Winking Unit 3-1 page 74

b.

d. c.

a.

12. FACTOR the following (not all may be factored):

a. x2 9x 18 b. 4062 xx c. x

2 5x14

d. 672 aa d. m28m16 e. 24112 gg

f. 652 xx g. 652 xx h. 6072 mm

i. 24142 2 gg j. xxx 60243 23 k. 234 3055 xxx


13. Special Forms

Name Formula Example Difference of two

squares BABABA 22 383838964 222 xxxx

Perfect square trinomials

222 2 BABABA

222 2 BABABA

2222 77724914 xxxxx

a. 362 x b. 92 m c. 814 m

d. 4004 2 b e. 9124 2 xx f. 94864 2 aa

g. 48 64121 ba h. mmm 324818 35 i.

6324 256036 yyxx

14. Find the volume of the rectangular prism

shown below

15. Describe the area of the shaded region as a

polynomial


15. Multiply the following:

a. 1223 xx b. 334 xx

16. FACTOR the following:

a. 216 2 xx b. 994 2 xx

c. 1572 2 xx d. 8103 2 aa

e. 8145 2 gg f. 24106 2 mm

g. bbb 30286 23 h. 12115 2 mm

M. Winking Unit 3-1 page 7ϋ

Unit 3 – 2 Name:

Solve the following QUADRATIC EQUATIONS using the SQUARE ROOT METHOD:

1. 0162 w 2. 22 48 0y 3.

24 196m

4. 3622b 5. 14213

2x 6.

31512

14

2

a

Solve the following QUADRATIC EQUATIONS by FACTORING & ZERO PRODUCT PROPERTY:

1. 2422 ww 2. 2082 tt 3. rr 652

4. 472022 xxx 5. xxx 24)4(2 6. xxx 1011352 22


(Continued) Solve the following QUADRATIC EQUATIONS by FACTORING & ZERO PRODUCT PROPERTY:

7. 01572 2 xx 8. pp 3104 2 9. xxxx 631223 22

10. 03212 23 xxx 11. 042 x 12. 08172 2 aa

Solve the applications that of QUADRATIC EQUATIONS:

1. The length of a rectangle is 1 cm more than

twice its width. If the area of the rectangle is

21cm2 then what are the dimensions?

2. The length of a rectangle is 3 cm less than

twice its width. If the area of the rectangle is

20cm2 then what are the dimensions?

3. The volume of the prism is 28 cubic inches.

What is the length of each side?

4. The volume of the prism is 45 cubic inches.

What is the length of each side?


5. A picture frame is shown at the right. If the

entire area of the frame and the picture totals

120 square inches find the width of the frame.

6. A below ground swimming pool is to be constructed

in the park. The pool is in the shape of a rectangle

with the dimensions of 20’ by 24’. A uniform width

sidewalk is to be made around the pool. If the

contractor says that he has enough concrete to create

300 ft2 of sidewalk. What is the maximum width of

the sidewalk around the pool?

7. The product of two consecutive positive integers

is 132. Write an equation to model the situation

and find the two integers.

8. The perimeter of a rectangle is 42 cm and the area

is 80 cm2. Write an equation to model the

situation and find the dimensions of the rectangle.

9. A park is putting in a sidewalk of uniform width to go around two sides of a rectangular garden that is

10 feet by 30 feet. The contractor has enough concrete for 176 ft2. What is the maximum width of such

a sidewalk?


10. A right triangle is shown below. Use the Pythagorean Theorem to determine the lengths of each side.

Notice – it is the

number that when added

to itself is equal to –5

and when multiplied by

itself is equal to .

BASIC

06102 xx 06102 xx Move constant to the other side

6102 xx 25625102 xx

3125102 xx Easily Factors

3155 xx Can be re-written

3152x Take the square root of both sides

3152x Don’t forget .

315 x Isolate x

315x OR 5678.105678.0 orx

INTERMEDIATE 23 15 9 0x x 09153 2 xx Move constant to the other side

9153 2 xx Divide both sides by the leading coefficients

3

9

3

153 2

xx

352 xx

2 25 3 255

4 1 4x x

4

25

4

12

2

5

2

5

xx

4

13

2

52

x

4

13

2

52

x

2

13

2

5x

2

13

2

5x

2

135x

3028.46972.0 orx

+6 +6

5

½ of

10 25 Square

5

–5 –5

–9 –9

2

5

½ of

5

Square

5/2

Add 25 to

both sides

Add 25 to

both sides

4

25


Solve the following by completing the square.

1. 019102 xx 2. 02122 xx

3. 2 12 8 0x x 4.

2 5 4 0x x

5. 01482 2 xx 6. 05183 2 xx


Solve the following by completing the square.

7. 22 16 3 0x x 8.

22 10 6 0x x

9. 22 5 12 0x x 10.

24 10 8 0x x


Solve the following by completing the square. Derive the quadratic formula.

02 cxbxa

a

acbbx

2

42


The Quadratic Formula Solve the following by quadratic formula

1. 019102 xx 2. 02122 xx

3. 08152 xx 4. 0472 xx

5. 01482 2 xx 6. 05183 2 xx

7. 03102 2 xx 8. 05213 2 xx

a

acbbx

2

42


Applications 1. Find the value of x that would make the diagram below accurate.

2. A golf ball is hit with an initial vertical velocity of 80 fps 216 80h t t

a. How high is the ball after 2 seconds?

b. How many seconds would it take the ball to hit the ground (the height would be h=0)?

c. When will the ball reach 48 feet?

d. What is the average rate of the change in height (i.e. vertical velocity) from t = 1 to t = 2 seconds? (Hint: Find the slope between the points (1, ___ ) and (2, ____ ). Evaluate the heights at t=1 and t=2 to find the corresponding coordinate)

e. For what values of t is the domain appropriate?

x+2

x x+4


3. A baseball his hit with an initial vertical velocity of 121 fps an the ball was struck 1 foot above ground.

112116 2 tth

a. How high is the ball after 2 seconds?

b. How many seconds would it take the ball to hit the ground (the height would be h = 0)?

c. When will the ball reach 30 feet?

d. When does the ball reach a maximum height?

e. What is the average vertical velocity from t = 0.2 to t = 0.9 seconds?

4. A person at a framing store is making a frame mat to go around a picture. The

mat is a uniform 2 inches around on each side. The picture’s width is 5 less

than twice the picture’s height. The entire area of the frame with picture

included is 221 square inches. What are the dimensions of the picture?


22.0 xy

22 xy

(REVIEW Unit 2-12) Name:

Consider the following EQUATIONS, make a table, plot the points, and graph what you think the graph looks like.

1. 2. 3. 4.

5. What happens to the graph as the number in front of x2 gets larger? Smaller?

6. 7. 8. 9.

10. What happens to the graph as the number in front of x2 is negative?

11. What happens when you add a number or subtract a number from x2?

x y

-4 -2 -1 0 1 2 4

x y

-3 -2 -1 0 1 2 3

x y

-2 -1.5 -1 0 1

1.5 2

x y

-1.5 -1

-0.5 0

0.5 1

1.5

x y

-3 -2 -1 0 1 2 3

x y

-2 -1.5 -1 0 1

1.5 2

x y

-3 -2 -1 0 1 2 3

x y

-3 -2 -1 0 1 2 3

22xy 2xy

25xy

25.0 xy 22xy 12 xy

M. Winking


2322 xy

13. 14. 15. 12.

16. What happens when you add a number or subtract a number from x inside the parenthesis?

17.What is a possible equation for the following graphs.

y = (x )2 y = (x )2 _

y = (x )2 y = (x )2 _

x y

-1 0 1 2 3 4 5

x y

-3 -2 -1 0 1 2 3

x y

-3 -2 -1 0 1 2 3

x y

-5 -4 -3 -2 -1 0 1

23 xy 22 xy 24 xy


18. Rewrite each of the following quadratics in vertex form by completing the square and graph.

a. 142 xxy b. 2 6 3y x x

c. 13122 2 xxy d. 22 16 27y x x

Domain:

Range:

Vertex: (Max or Min)

Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:


(18 continued) Rewrite each of the following quadratics in vertex form by completing the square and graph.

e. 2 7 6y x x f.

2 11 24y x x

g. 23 15 10y x x h.

22 15 22y x x

Domain:

Range:


Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:

Domain:

Range:


Axis of Symmetry:


Name:

1. Consider the function: 2 6f x x x

a. What are the zeros of the function (using factoring)?

b. What is the axis of symmetry of the parabola?

c. What is the vertex of the parabola (graph the parabola)?

2. Consider the function: 2 8 15f x x x




3. Consider the function: 22 3f x x x





23. The expression 2 70 600P x x represents a company’s profit for selling x items.

a. What are the break-even point(s) for selling x items (i.e. how many items sold yields a profit of 0?)

b. Is the vertex a minimum or maximum?

c. If the model is accurate, how many items should the company sell to maximize their profit and what is the maximum profit?

24. The expression 216 400 5h t t represents the height of a cannonball t seconds after it was fired. What is the

maximum height of the cannon ball and how many seconds did it take to reach its maximum height?

25. The expression 2 44 490C x x represents the cost in $1000 of dollars per year that company must spend out of

pocket on each employee for health insurance for x number of employees. How many employees should the company hire

to minimize their cost of health insurance?


Name:

1. Solve the following quadratic inequalities in one variable:

a. 01032 xx b. 2 1 12 0x x

c. 22 2 24x x d. 22 14 20x x

e. 23 2 8 0x x f.

22 13 24 0x x


Name:

1. Given the formula for the perimeter of a rectangle is:

𝟐𝒍 + 𝟐𝒘 = 𝑷

rewrite the formula so that it has been solved for the variable ‘w’.

2. Given the formula for Power in Watts of an electrical circuit is:

𝑷 = 𝑰 ∙ 𝑽

where I is the resistance in Ohms and V is the voltage in volts, rewrite the formula so that

it has been solved for ‘V’.

3. Given the formula for Centripetal Acceleration can be described by the formula:

𝒂 =𝒗𝟐

𝒓

where ‘v’ is the velocity in meters per second (m/s) and ‘r’´ is the length of the radius in meters.

Assuming all variables represent positive values, rewrite the formula so that it has been solved for ‘v’.

4. Given the formula for the area of a trapezoid is:

𝒉

𝟐(𝒃𝟏 + 𝒃𝟐) = 𝑷

Rewrite the formula so that it has been solved for the variable ‘b1’.


5. Given the function, 𝒇(𝒙) = 𝒙𝟐 + 𝟐𝒙,

determine the average rate of change from

x = 1 to x = 2.

6. Given the function, 𝒑(𝒙) = 𝟐𝒙 + 𝟏,

determine the average rate of change from

x = 0 to x = 2.

7. Given the table of values for 𝒉(𝒙), x – 2 0 2 4 6

h(x) 3 – 1 3 15 35

What is the average rate of change

from x = 2 to x = 6?

8. Given the table of values for 𝒈(𝒙), x – 2 0 2 4 6

g(x) 3 – 1 3 15 35

What is the average rate of change from

x = – 2 to x = 4?

9. Given the graph of q(x),

what is the average rate of change from

x = – 1 to x = 3.

10. Given the graph of q(x),

what is the average rate of change from

x = – 1 to x = 3.


11. A group of students visited Stone Mountain. They decided

to walk up to the top of the mountain. At 3:00 pm they

started walking and according to their GPS when they were

at the bottom of the mountain their elevation was 861 feet

above sea level. At 3:45 pm they were at the top of the

mountain which was 1686 feet above sea level. What is the

students’ average rate of change in feet per minute?

12. A college student is driving from Athens to Panama City

Beach for a vacation. The student left Athens at 12:00pm

and arrived at the beach at 5:15pm but gained an hour due

to the Standard Time Zone change. The trip was exactly

350miles. What was the student’s average rate of speed in

miles per hour?

Do you think the student ever traveled more than 58 miles per hour?


Name:

1. To rent a bicycle on the beach board walk, the initial cost is $10 plus an additional charge of $5 per hour. Create a linear function model and graph.

2. Let h(x) be the number of person-hours it takes to assemble x engines in a factory. The company’s accountant determines that the time it takes depends on start-up time and the number of engines to be completed. It takes 6.5 hours to set up the machinery to make the engines and about 5.25 hours to completely assemble one. Create the function h(x).

3. Two students were selling cookies for a fundraiser. They both started on

the first day of the month. The graph shows how many boxes Cindy sold in total after each day. If Veronica hadn’t sold any boxes by the 3rd day of the month but then started selling cookies at the exact same rate, how many cookie boxes will Veronica have sold by the 12th day of the month?

4. A babysitter charges parents $10 for coming to their house and then $6

for every hour she babysits. Create function c(x), where x = the number of hours and c(x) represents the amount the babysitter charges.


5. The function graphed shows the decibel level a car horn makes at varying distances away from the horn. a. At prolonged period of listening anything above 90

decibels can cause permanent ear damage. How far must you be away from the horn to prevent potential ear damage?

b. How many decibels would the car horn be at 70 feet away?

6. Bonnie is renting space at a local private school to offer math tutoring. The school charges her a

monthly fee to use classrooms after school and she charges the students a flat rate per hour. She models her profit using the function: 𝑝(𝑥) = 𝑐 ∙ 𝑥 − 𝑘. Explain what you think the variable x and the parameters ‘c’ and ‘k’ should each represent.

7. A pair of shoes cost $135 and has been discounted 38%. What is the sale price of the shoes?

8. A person paid their electricity bill late.

The bill is $78. The electric company charges an 12% late charge. What is the total cost of the bill after paying it late?

9. If a store has marked down all of its

products 20%, create a function that would represent the new cost after the discount, d(x), where x is the original price of the item before the discount.

10. If a furniture store buys furniture at wholesale cost of x and marks up the furniture by 95% to sell on their showroom floor, create a function p(x) that represents the price of the item in the store.

11. A manager at a jewelry store buys jewelry at whole sale cost of x. To set a price on the item in the store she first adds $25 and she calls this function a(x). Then, she increases by 80% and calls this combined

price increase 𝑝(𝑎(𝑥)). Create a function model that expresses the overall increase in price.


12. Describe each of the following as examples of relationships that could be modeled by: LINEAER functions, QUADRATIC functions, EXPONENTIAL functions, or NONE OF THESE.


13. Describe each of the following as examples of relationships that could be modeled by: LINEAER functions, QUADRATIC functions, EXPONENTIAL functions, or NONE OF THESE.

The Sequence: {4, 12, 36, 108, 324,… } The Sequence: {3, 9, 15, 21, 27, 33,… }


1....Unit 3 – 2 Name: Solve the following QUADRATIC EQUATIONS using the SQUARE ROOT METHOD: 1. w2...

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Transcript of 1....Unit 3 – 2 Name: Solve the following QUADRATIC EQUATIONS using the SQUARE ROOT METHOD: 1. w2...