Cache County School District 2013-2014
Secondary II Math
Utah Integrated
Mathematics Core
Student Edition - Honors
Unit 5: Solving
and Exploring
Quadratics
Secondary II Unit 5 – Solving and Exploring Quadratics: Table of
Contents
Homework Help (QR Codes and links to videos, tutorials, examples)………………….
Section 5.1 –Solving by Factoring, Teacher Notes…………………………………………
Notes, Assignment……………………………………………………………………
Section 5.2 Completing the Square Task, Teacher Notes ………………………………………………….
Notes, Assignment……………………………………………………………………
Section 5.3 – Simplifying Radicals Review, Teacher Notes ..................................................
Notes, Assignment ........................................................................................................
Section 5.4 – Deriving the Quadratic Formula Task .............................................................
Quadratic Formula with Rational Solutions Notes, Assignment………………
Section 5.5 – Quadratic Formula with Irrational Solutions Task…………………………
Teacher Notes, Notes, Assignment ................................................................................
Section 5.6 – Discovering the Discriminant Task…………………………………………….
Quad Formula with Complex Solutions, Teacher Notes……………………….
Notes, Assignment ................................................................................................... .
Section 5.7 – Perfecting my Quads Task, Teacher Notes………………………………...
Assignment……………………………………………………………………..
Practice Worksheet: Solving Quadratic Equations……………………………………
Section 5.8- Building Functions That Model Relationships Task…………………………...
Teacher Notes, Notes, Assignments………………………………………………….
Section 5.9 – Graphing Parabolas by Plotting Points Task,……………………………………
Teacher Notes, Notes, Assignment……………………………………………………
Secondary II Unit 5 - Exploring Quadratics: Homework Help
video
Section 5.1
http://goo.gl/VlO82
http://goo.gl/ymULA
http://goo.gl/ujIq5
Section 5.2
http://goo.gl/EiCzy
http://goo.gl/Tp3A0
http://goo.gl/yGFPU
Section 5.3
http://goo.gl/FjcbL
http://goo.gl/TYN6
http://goo.gl/vnFw0
Section 5.4
video
http://goo.gl/YvhFv
http://goo.gl/wHarz
http://goo.gl/fMfBL
http://goo.gl/pDRr9
http://goo.gl/fMfBL
Section 5.5
http://goo.gl/hdoU3
video
Section 5.6
http://goo.gl/BZdXu
http://goo.gl/MyYPZ
http://goo.gl/jNZ3f
Section 5.7, 5.8, 5.9 – No additional resources for this section.
See www.cachemath2.wordpress.com for other homework help.
Unit 5 Lesson 1 – Solving by Factoring
Notes 5.1
Vocabulary
Quadratic Equation:
The Zero Factor Property:
Solutions to quadratic equations:
Question: What about prime quadratic polynomials? Can you solve them by factoring?
Answer: No. You can’t factor a polynomial that is prime. You can solve them, but not by factoring. We
will learn more methods later on.
Example 1
Solve the equation (𝑥 + 4)(𝑥 − 3) = 0
Example 2
Solve the equation 3𝑥 2 = −3𝑥.
Solve by factoring:
Solve by dividing by 3x and taking the square
root.
Check your solutions to the equation 𝟑𝒙𝟐 = −𝟑𝒙.
CAUTION! If in Example 1 you divide each side of the equation 3𝑥 2 = −3𝑥 by 3x, you would get 𝑥 =
−1 but not the solution𝑥 = 0. For this reason we usually do not divide each side of an equation by a
variable.
Example 3
Solve 𝑥 2 + 𝑥 − 6 = 0
Example 4
Example 5
Solve (2𝑥 + 1)(𝑥 − 1) = 14
Solve 5𝑥 2 − 30𝑥 + 45 = 0
Example 6
Example 7
Solve 2𝑥 3 − 𝑥 2 − 8𝑥 + 4 = 0
Solve ℎ2 = −5ℎ
Example 8
Solve 3𝑥(2𝑥 + 1) = 18
Unit 5 Lesson 1 – Solving by Factoring
Ready, Set, Go! - Assignment 5.1
Name______________________________
Date_________ Hour_______
http://goo.gl/VlO82
Ready
1. Decide if the given statement is true or false. Write your answer on the given line.
a.
The equation 𝑥(𝑥 + 2) = 3 is equivalent to 𝑥 = 3 𝑜𝑟 𝑥 + 2 = 3. _______________________
b.
Equations solved by factoring always have two different solutions. ______________________
c.
The equation 𝑎 ∙ 𝑑 = 0 is equivalent to 𝑎 = 0 𝑜𝑟 𝑑 = 0. ________________________
d.
Both 1 𝑎𝑛𝑑 − 4 are solutions to the equation (𝑥 − 1)(𝑥 + 4) = 0. ______________________
e.
The solutions to 3(𝑥 − 2)(𝑥 + 5) = 0 𝑎𝑟𝑒 3, 2, 𝑎𝑛𝑑 − 5. _________________________
f.
If x is the width in feet of a rectangular room and the length is 5 feet longer than the width, then
the area is 𝑥 2 + 5𝑥 square feet. ___________________________
2. Why don’t we usually divide each side of an equation by a variable?
3. What is the zero factor property? Explain in your own words.
4. What is a quadratic equation?
Set
Solve each equation. Write your answers as a solution set.
5.
(𝑎 + 6)(𝑎 + 5) = 0
6.
3(3𝑘 − 8)(4𝑥 + 3) = 0
7.
𝑥 2 + 3𝑥 + 2 = 0
8.
𝑤 2 − 9𝑤 + 14 = 0
9.
2𝑚2 + 𝑚 − 1 = 0
10.
𝑚2 = −7𝑚
11.
3𝑥 2 − 10𝑥 = −7
12.
(𝑥 + 2)(𝑥 − 6) = 20
13.
2(4 − 5ℎ) = 3ℎ2
14.
2𝑤(4𝑤 + 1) = 1
15.
4𝑚2 − 12𝑚 + 9 = 0
16.
𝑛3 − 3𝑛2 + 3 = 𝑛
Go!
11
17.
𝑥 3 = 4𝑥
18.
𝑧2 +
19.
(𝑥 − 2)2 + 𝑥 2 = 10
20.
𝑚4 + 𝑚3 = 100𝑚2 + 100𝑚
21.
16𝑥 2 + 8𝑥 + 1 = 0
22.
5𝑎3 = 45𝑎
23.
(2𝑥 − 1)(𝑥 2 − 9) = 0
24.
(3𝑥 − 5)(25𝑥 2 − 4) = 0
25.
𝑛3 − 2𝑛2 − 𝑛 + 2 = 0
2
𝑧 = −6
Unit 5 Lesson 2 – Completing the Square
Task 5.2
Name______________________________
Date_________ Hour_______
In this activity, you will complete the square using algebra tiles.
To illustrate this method, let’s complete 𝑥 2 − 6𝑥 to a perfect square by adding a constant. Ask your
teacher how they want you do show negative x values. You can always flip your cards over to show the
negatives if no other directions are given.
The model for the given expression 𝑥 2 − 6𝑥 is as shown to the right.
To “complete the square” you need 9 small unit square tiles. Add these
the array of tiles. You now have a model for the expression
𝑥 2 − 6𝑥 + 9 which is the completed square (𝑥 − 3)2 .
x2
–x –x –x
to
–x
–x
–x
Part 1: Your Turn
1.
To use this method outlined above for the expression 𝑥 2 + 10𝑥, how many large square tiles do you
use? __________________________
How many long rectangular tiles do you use? _____________________
Are they positive or negative? ___________________
2.
How do you split up the long rectangular tiles? ___________________________
3.
How many small square tiles do you need to complete the square? ______________________
4.
The complete square is _________________________.
Complete the square, using algebra tiles as described above. Draw a sketch
5. 𝑥 2 − 14𝑥 ___________
6.
𝑥 2 + 12𝑥 __________
7.
𝑥 2 + 8𝑥 ____________
8.
𝑥 2 − 16𝑥 __________
9.
𝑥 2 − 18𝑥 ___________
10.
𝑥 2 + 2𝑥 __________
Part 2:
Before we can solve a quadratic equation by completing the square, we must see what it takes to be a
perfect square trinomial.
Example 1: Which of the following are perfect square trinomials?
a.) x 2  6 x  9
b.) x 2  6 x  36
c.) x 2  12 x  36
2
b
CONCLUSION: When a trinomial is of the form x  bx  c , it is a perfect square if c    . In this
 2
2
b

case we get x  bx  c   x  
2

2
2
“a perfect square”.
Example 2: Determine the constant that should be added to make each expression a perfect square.
trinomial:
a.) x 2  10 x
b.) x 2  24 x
c.) x 2 
3
x
2
d.) x 2  7 x
Example 3: Solve x 2  8 x  4  0 by completing the square.
We know that we complete the square on the x 2  8 x terms. So isolate these terms.
x 2  8 x  4
Now complete the square:
x 2  8 x  ____  4  ____
x  __   ____
This gives
Now use the square root method to solve:
2
x  ___ 2

x  ___  
Simplify the radical:
x  ___  
Now solve for x .
x  ____ 
Example 4: Solve 2 x 2  3  5 x by completing the square.
Isolate the x and x 2 terms.
2 x 2  5x  3
We need the coefficient of x 2 to be 1. So divide both sides by 2.
5
3
x2  x 
2
2
Now complete the square:
5
3
x 2  x  ____   ____
2
2
2
x  __   ____
This gives
Now use the square root method to solve:

 x 

Simplify the radical:
2

  

x  ___   _____
Now solve for x .
x  ____ ______  x  ___ or x  _____ .
Unit 5 Lesson 2 – Completing the Square
Notes 5.2
Solve each quadratic equation by completing the square.
a.) x 2  4 x  3
b.) y 2  2 y  5  0
c.) x 2  1  x
d.) 4 y 2  12 y  7  0
e.) 2 x 2  5 x  4  0
f.) 4 x 2  12 x  9  100
Unit 5 Lesson 2 – Solving by Completing the Square
Ready, Set, Go! - Assignment 5.2
Name______________________________
Date_________ Hour_______
http://goo.gl/EiCzy
Ready
Solve each equation by the even-root property.
9
1.
𝑥 2 = 81
2.
𝑥2 = 4
4.
(𝑥 − 3)2 = 16
5. (𝑥 + 5)2 = 4
Find the perfect square trinomial whose first two terms are given.
7. 𝑥 2 + 2𝑥
8. 𝑚2 + 14𝑚
1
10. 𝑤 2 + 4 𝑤
3
11. 𝑧 2 + 2 𝑧
3 2
9. 𝑥 2 − 3𝑥
12. 𝑤 2 − 5𝑤
16.
7
6. (𝑤 − 2) = 4
Set
Solve by completing the square. Use a calculator to check your answers.
13. 𝑥 2 − 2𝑥 − 15 = 0
14.
𝑥 2 − 6𝑥 − 7 = 0
15. 2𝑥 2 − 4𝑥 = 70
𝑥 2 = 32
3.
3𝑥 2 − 6𝑥 = 24
17. 𝑦 2 − 3𝑦 − 10 = 0
18.
2ℎ2 − ℎ − 3 = 0
19. 2𝑚2 − 𝑚 − 15 = 0
20.
𝑞 2 + 5𝑞 = 14
21. 𝑥 2 + 6𝑥 + 5 = 0
22.
2𝑥 2 + 3𝑥 − 2 = 0
23. 𝑥 2 − 3𝑥 − 6 = 0
.
Unit 5 Lesson 3 – Simplifying Radicals Review
In-Class Guided Notes 5.3
Vocabulary:



√25𝑎2 𝑏
Radical
Radicand
Index
Fill in the following chart: Follow the same pattern until all boxes are completed.
12 =
22 =
32 =
√1 =
√4
13 =
23 =
33 =
3
√1 =
√8 =
3
Radicals may also contain variables. We will learn how to simplify those as well.
√𝑎2 𝑏 4 𝑐 2
√𝑥 3 𝑦 6 𝑧 5
√4𝑤 5 𝑦 3 𝑧12
3
5
√𝑎10 𝑏15 𝑐14 𝑑100
3
3
3
√27𝑥 3 𝑦 9 𝑧13
√𝑎3 𝑏 6 𝑐 11 𝑑 22
√3𝑎3 𝑏 3
√8𝑎15
Simplify.
Example 1:
72
72
2.
28
=
_____________
3.
45
=
_____________
4.
24
=
_____________
5.
300x3 y 5 =
_____________
6.
128a3b11
8.
6 108x11 y13
=
7. 5 75a2b2 =
_____________
= _____________
9. 10 275
=
10.
3
24
11.
3
12.
3
128a3b
13.
4 3 250a2b5c
81
_____________
_____________
Unit 5 Lesson 3 – Simplifying Radicals Review
Ready, Set, Go! - Assignment 5.3
Name______________________________
Date_________ Hour_______
http://goo.gl/FjcbL
3
1. Explain in words how you would simplify the problem √𝑎3 𝑏 4 𝑐 5 𝑑15 . Your answer should include the
process taken, not just the final answer.
2. Complete the following table.
12 =
22 =
32 =
42 =
52 =
62 =
72 =
13 =
23 =
33 =
43 =
53 =
63 =
73 =
3. Find the “?” in each of the following problems. (State what the “?” would have to be to make each
statement true)
2
√64 =?
3
√8 =?
2√100 =?
3
√? = 81
2(√? ) = 8
3
√? 𝑥 4 = 7𝑥 2
√64 =?
√1000𝑥 ? = 10𝑥
Simplify each radical expression
4.
√4𝑎2 𝑏 4 𝑐 7
5.
6.
√25𝑒 2 𝑓 5 𝑔
7.
3
√27𝑥 11 𝑦
2
√16𝑘14 ℎ12 𝑟 2
3
√125𝑥 2 𝑦 3 =?
(? )3 = 216
8.
7
√𝑥 4 𝑦 7 𝑧11
9.
10.
√𝑓 3 𝑔22 ℎ30
11.
3
√𝑢𝑣𝑤
5
√𝑥10 𝑦 40 𝑧 60
12.
28
=
_____________
13.
44
=
_____________
14.
45
=
_____________
15.
24
=
_____________
16.
300 =
_____________
17.
128 =
_____________
18.
75
=
_____________
19.
288 =
_____________
20.
108 =
_____________
21.
275 =
_____________
22.
99
=
_____________
23.
490 =
_____________
24.
32
=
_____________
25.
20
_____________
26. True or False:
4
=
24 = 16 𝑎𝑛𝑑 42 = 16. 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒, √16 =
Unit 5 Lesson 4 – Deriving the Quadratic Formula
Task 5.4
Name_________________________________
Date____________ Hour_____
Directions:
Working in pairs or groups solve the equation 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 by completing the square. Show each
step of your work.
Unit 5 Lesson 4 – Quadratic Formula with Rational Solutions
Notes 5.4


This formula can be used to solve ANY quadratic equation of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
What is the quadratic formula?
Two Rational Solutions:
Solve 𝒙𝟐 − 𝟏𝟐𝒙 = 𝟐𝟖 by using the Quadratic Formula
Solve 𝒏𝟐 − 𝟐𝒏 − 𝟒𝟖 = 𝟎 by using the Quadratic Formula.
Solve 𝟐𝒑𝟐 + 𝒑 − 𝟑𝟔 = 𝟎 by using the Quadratic Formula.
Solve 𝟑𝒎𝟐 = 𝟕𝟓 by using the Quadratic Formula.
Solve 𝟔𝒃𝟐 − 𝟏𝟎𝒃 − 𝟐 = 𝟐 by using the Quadratic Formula.
Unit 5 Lesson 4 – Quadratic Formula with Rational Roots
Ready, Set, Go! - Assignment 5.4
Name_____________________________________
Date_________ Hour_______
http://goo.gl/YvhFv
Ready
1. Decide if each statement is either true or false.
a.
Completing the square can be used to develop the quadratic formula ____________________.
b.
The quadratic formula can be used to solve any quadratic equation in one variable
_______________.
c.
Completing the square can be used to solve any quadratic equation in one variable
_______________.
d.
Factoring can be used to solve any quadratic equation in one variable. _________________.
e.
For the equation 3𝑥 2 = 4𝑥 − 7, we have 𝑎 = 3, 𝑏 = 4, 𝑎𝑛𝑑 𝑐 = −7 ___________________.
f.
If 𝑎 = 2, 𝑏 = −3, 𝑎𝑛𝑑 𝑐 = −4, 𝑡ℎ𝑒𝑛 𝑏 2 − 4𝑎𝑐 = 41 _________________.
Set
Solve each equation with the quadratic formula.
2.
𝑏 2 = 16
3.
𝑥 2 − 3𝑥 = 0
4.
2𝑟 2 − 32 = 0
5.
5𝑥 2 − 2𝑥 − 3 = 0
6.
3𝑥 2 − 9𝑥 − 54 = 0
7.
𝑘2 − 1 = 0
Go!
Solve each equation with the quadratic formula.
8.
2𝑝2 − 7𝑝 − 4 = 0
9
6𝑥 2 − 30 = −6
10
4𝑝2 = −6𝑝 + 40
11.
3𝑥 2 − 95 = −4𝑥
12.
𝑚2 − 48 = 2𝑚
13.
3𝑣 2 = 27
14.
2𝑥 2 − 4 = −7𝑥
15.
𝑛2 = 2𝑛 + 24
Unit 5 Lesson 5 – Quadratic Formula with Irrational Solutions
Task 5.5
Name_______________________________
Date___________ Hour______
Part 1
Directions: Solve the equation 5𝑥 2 = 125 by factoring, the even root property, completing the square,
and the quadratic formula. Justify each method using mathematical properties.
Even – Root Property
Factoring
Completing the Square
The Quadratic Formula
Question 1:
For this particular problem 5𝑥 2 = 125, which method did you prefer? Justify your opinion with a few
sentences. (your opinion does not necessarily need to be the same as your partner’s or group’s).
Question 2:
Do you think this method would be your method of choice no matter the quadratic equation? Why or why
not. Rationalize your answer with a few sentences and examples.
Part 2
Solve the equation 6𝑥 2 − 𝑥 − 15 = 0 by factoring, completing the square, and the quadratic formula.
Factoring
Completing the Square
The Quadratic Formula
Question 3:
For this particular problem 6𝑥 2 − 𝑥 − 15 = 0, which method did you prefer? Justify your opinion with a
few sentences. (your opinion does not necessarily need to be the same as your partner’s or group’s).
Question 4:
Do you think this method would be your method of choice no matter the quadratic equation? Why or why
not. Rationalize your answer with a few sentences and examples.
Question 5:
Was your favorite method in Part 1 the same as Part 2? Why or Why not?
Unit 5 Lesson 5 – Quadratic Formula with Irrational Solutions
Notes/In-Class Examples 5.5
Two Irrational Solutions:
Example 1: 𝑥 2 + 𝑥 − 7 = 0
Example 2:
2𝑥 2 + 2𝑥 − 2 = 0
Example 3:
𝑛2 = 4 − 2𝑛
Example 4:
4𝑛2 − 9 = 6
Example 5:
10𝑘 2 − 𝑘 − 11 = −3
Example 6:
2𝑥 2 + 6𝑥 + 3 = 0
Unit 5 Lesson 5 – Quadratic Formula with Irrational Solutions
Ready, Set, Go! - Assignment 5.5
Name_________________________________
Date_________ Hour_______
http://goo.gl/hdoU3
Solve each quadratic function using the QUADRATIC FORMULA. Write your answer in the simplified radical
form, and then give your answers as a decimal approximation to two decimal places in the space provided. Show
your work on a separate piece of paper.
1.
𝑥 2 + 5𝑥 + 5 = 0
2.
2𝑥 2 − 4𝑥 + 1 = 0
3.
𝑥 2 − 6𝑥 − 2 = 0
5. 2𝑥 2 + 4𝑥 − 5 = 0
6. 9𝑥 2 = 4 + 7𝑛
7. 𝑥 2 = 𝑥 + 3
9. 2𝑥 2 + 39 = −18𝑥
10. 3𝑥 2 + 2𝑥 = 2
11. 𝑥 2 − 4𝑥 − 10 = 0
4. 4𝑥 2 − 1 = −8𝑥
8. 2𝑥 2 + 23 = 14𝑥
Answers: (Write your solutions below in both radical and decimal form. See Directions)
1.
Radical Form
Decimal Approximation


2.
Radical Form
Decimal Approximation


3.
Radical Form
Decimal Approximation


4.
Radical Form
Decimal Approximation


5.
Radical Form
Decimal Approximation


6.
Radical Form
Decimal Approximation


7.
Radical Form
Decimal Approximation


8.
Radical Form
Decimal Approximation


9.
Radical Form
Decimal Approximation


10.
Radical Form
Decimal Approximation


11.
Radical Form
Decimal Approximation


Unit 5 Lesson 6 – Discovering the Discriminant
Task 5.6
Name________________________________
Date______________ Hour_____
Directions: For this multiple choice section - In your group, or with your table partner, answer the
following questions. It may be helpful to use your notes from this unit.
1.
What are possible solutions for a quadratic equation with −6 as a discriminant?
𝑎. √6, −√6
b. −5, 3
c.
𝑖√5
2
,−
𝑖√5
d. −6
2
Justify your choice.
2.
What are possible solutions for a quadratic equation with 36 as a discriminant?
𝑎.
√2
√2
,− 5
5
b.
−7, 3
𝑐.
𝑖√2
5
, −
𝑖√2
d. 4
5
Justify your choice.
3.
What are possible solutions for a quadratic equation with 5 as a discriminant?
a.
√5, −√5
b.
9, −23
c.
3𝑖√5
2
,
−
3𝑖√5
2
d. 11
Justify your choice.
4.
What are possible solutions for a quadratic equation with 0 as a discriminant?
a. √21, −√21
Justify your choice.
b. −2, −24
c.
4𝑖√3
11
,−
4𝑖√3
11
d. 1
Fill in the following chart:
Value of Discriminant
Number of Roots
Type of Roots
64
-8
21
0
32
-36
25
On each example:
a. Find the discriminant. b. State which types of roots the problem will have. c. Then, use the
quadratic formula to find the exact roots.
Example 1:
𝑥 2 + 4𝑥 + 4 = 0
Example 2:
9𝑥 2 + 12𝑥 + 4=0
Unit 5 Lesson 6 – Quadratic Formula with Complex Solutions
Notes 5.6
Solve each example using the quadratic formula. Write your answers in standard form.
Example 1:
9𝑥 2 − 6𝑥 − 3 = −8
Example 2:
4𝑥 2 + 𝑥 + 1 = 0
Example 3:
𝑥(3𝑥 + 4) = −2
Unit 5 Lesson 6 – Quadratic Formula with Complex Solutions
Ready, Set, Go! - Assignment 5.6
Name______________________________
http://goo.gl/BZdXu
Date_________ Hour_______
Ready
1. Fill in the following table.
1.
2.
3.
4.
5.
6.
7.
2.
Value of Discriminant
118
0
-196
256
12
0
225
Type of Roots
169
What are possible solutions for a quadratic equation with
a. √13, −√13
3.
Number of Roots
b. −2, 5
c.
4𝑖√5
7
,−
4𝑖√5
7
as a discriminant?
13
d.
What are possible solutions for a quadratic equation with −15 as a discriminant?
a.
3+𝑖√15 3−𝑖√15
2
,
2
b.
4, −9
c.
√15
√15
,
−
5
5
d.
5
Set
A) Find the value of each discriminant. B) Determine how many and what type of roots by looking
at the discriminant. C) Prove B by using the Quadratic Equation to find the solutions.
4.
5𝑥 2 − 6 = 0
A. ______________________
B. ______________________
C. ______________________
5.
𝑥 2 = −49
A. ______________________
B. ______________________
C. ______________________
6.
2𝑥 2 − 3𝑥 = −2
A. ______________________
B. ______________________
C. ______________________
7.
2𝑥 2 − 5𝑥 = 6
A. ______________________
B. ______________________
C. ______________________
8.
8𝑥 − 1 = 4𝑥 2
A. ______________________
B. ______________________
C. ______________________
Go!
Solve each quadratic equation by using the quadratic formula. Be sure to write your answers in standard
form.
9.
𝑥 2 − 3𝑥 + 6 = 0
10.
𝑥 2 − 5𝑥 + 20 = 0
11.
9𝑥 2 − 6𝑥 = −7
12.
𝑡 2 + 11 = −4𝑡
13.
𝑧(2𝑧 + 3) = −2
14.
(2𝑥 − 1)(8𝑥 − 4) = −1
Unit 5 Lesson 7 – Perfecting my Quads
Task 5.7
Name_________________________________________
Date___________ Hour___________
Carlos and Clarita, Tia and Tehani, and their best friend Zac are all discussing their favorite methods for
solving quadratic equations of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0. Each student thinks abou the related quadratic
function 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 as part of his or her strategy.
Carlos: “I like to make a table of values for x and find the solutions by inspecting the table.”
Clarita: “I like to write the equation in factored form, and then use the factors to find the
solutions.”
Tia: “I like to treat it like a quadratic function that I am trying to put in vertex form by completing
the square. I can then use a square root to undo the squared expression.”
Tehani: “I also like to treat it like a quadratic function, but I use the quadratic formula to find the
solutions.”
Zac: “I like to graph the related quadratic function and use my graph to find the solutions.”
Demonstrate how each student might solve each of the following quadratic equations.
1. Solve:
Carlos’ Strategy
Zac’s Strategy
𝑥 2 − 2𝑥 − 15 = 0
Clarita’s Strategy
Tia’s Strategy
Tehani’s Strategy
2. Solve:
Carlos’ Strategy
Zac’s Strategy
Clarita’s Strategy
Tia’s Strategy
Tehani’s Strategy
3. Solve:
Carlos’ Strategy
Zac’s Strategy
Tia’s Strategy
Tehani’s Strategy
2x 2 + 5x − 12=0
𝑥 2 + 4𝑥 − 8 = 0
Clarita’s Strategy
4. Solve:
Carlos’ Strategy
Zac’s Strategy
Tia’s Strategy
Tehani’s Strategy
8𝑥 2 + 2𝑥 = 3
Clarita’s Strategy
Describe why each strategy works.
As the students continue to try out their strategies, they notice that sometimes one strategy works better
than another. Explain how you would decide when to use each strategy.
Unit 5 Lesson 7 – Perfecting my Quads
Ready, Set, Go! - Assignment 5.7
Name_________________________________________
Date___________ Hour___________
Ready
The given functions provide the connection between possible areas, A(x), that can be created by a
rectangle for a given side length, x, and a set amount of perimeter. You could think of it as the different
amounts of area you can close in with a given amount of fencing as long as you always create a
rectangular closure.
1. 𝐴(𝑥) = 𝑥(10 − 𝑥)
2. 𝐴(𝑥) = 𝑥(50 − 𝑥)
Find the following:
𝐴(3)
𝐴(4)
Find the following:
𝐴(10)
𝐴(20)
𝐴(6)
𝐴(𝑥) = 0
𝐴(30)
𝐴(𝑥) = 0
𝑊ℎ𝑒𝑛 𝑖𝑠 𝐴(𝑥)𝑎𝑡 𝑖𝑡𝑠 𝑚𝑎𝑥𝑖𝑢𝑚𝑢𝑚?
𝐸𝑥𝑝𝑙𝑎𝑖𝑛 𝑜𝑟 𝑠ℎ𝑜𝑤 ℎ𝑜𝑤 𝑦𝑜𝑢
𝑘𝑛𝑜𝑤.
𝑊ℎ𝑒𝑛 𝑖𝑠 𝐴(𝑥)𝑎𝑡 𝑖𝑡𝑠 𝑚𝑎𝑥𝑖𝑢𝑚𝑢𝑚?
𝐸𝑥𝑝𝑙𝑎𝑖𝑛 𝑜𝑟 𝑠ℎ𝑜𝑤 ℎ𝑜𝑤 𝑦𝑜𝑢
𝑘𝑛𝑜𝑤.
Set
For each of the given quadratic equations find the solutions using an efficient method. State the method
you are using as well as the solutions. You must use at least three different methods on the following 6
problems.
3.
𝑥 2 + 17𝑥 + 60 = 0
4.
𝑥 2 + 16𝑥 + 39 = 0
5.
𝑥 2 + 7𝑥 − 5 = 0
6.
3𝑥 2 + 14𝑥 − 5 = 0
7.
𝑥 2 − 12𝑥 = −8
8.
𝑥 2 + 6𝑥 = 7
Go!
Write each of the quadratic expressions in factored form.
9.
𝑥 2 + 3𝑥 + 2
10.
2𝑥 2 + 3𝑥 + 1
11.
𝑥 2 − 3𝑥 + 2
12.
𝑥 2 + 𝑥 − 12
13.
𝑥 2 + 8𝑥 − 20
14.
𝑥 2 − 5𝑥 − 6
Practice Worksheet: Solving Quadratic Equations
Name_________________________________________
Date___________ Hour___________
Solve each quadratic equation by the “square root” method, factoring, completing the square, or
the quadratic formula. Remember that you can use a graphing calculator to check your answers.
1. 4(𝑥 − 2)2 + 2 = 326
2. 13 − 8𝑛2 = −1139
3. (𝑛 + 2)(2𝑛 + 5) = 0
4. 9𝑟 2 − 5 = 607
5.
10𝑥 2 + 29𝑥 + 10 = 0
6.
𝑥 2 − 10𝑥 = −9
7. 18𝑥 2 + 10𝑥 = −3𝑥 + 21
8. 𝑥 2 + 6𝑥 = 5
9. 𝑥 2 + 5𝑥 − 2 = 0
10. −2𝑥 2 + 3𝑥 + 2 = −2𝑥 − 1
11. 6𝑥 2 − 5𝑥 − 13 = 𝑥 2 − 11
12. 2(𝑥 − 3)2 + 2𝑥 − 9 = 0
13. 2(𝑥 2 − 5) = −𝑥 2 − 1
14.
15. 6𝑥 2 − 5𝑥 − 13 = 𝑥 2 − 11
2(𝑥 − 3)2 = −2𝑥 + 9
Unit 5 Lesson 8 –Building Functions that Model Relationships
Task 5.8
Name__________________________________________
Date_______ Hour_____
Vocabulary (much of this vocabulary is covered in Secondary Math 1)
A function is a relation for which each input has exactly one output. In an ordered pair the first number
is considered the input and the second number is considered the output. If any input has more than
one output, then the relation is not a function.
For example the set of ordered pairs {(1,2), (3,5), (8,11)} is a function because each input value has
an output value. The set {(1, 2) (1, 3), (6, 7)} does not represent a function because the input 1 has
two different outputs 2 and 3.
Linear Function: a function that can be written in the form 𝑦 = 𝑚𝑥 + 𝑏, where
m and b are constants. The graph of a linear function is a line.
A linear function can be expressed in two different ways:
Linear notation: 𝑦 = 𝑚𝑥 + 𝑏
Function notation: 𝑓(𝑥) = 𝑚𝑥 + 𝑏
Linear functions can model arithmetic sequences, where the domain is the set
of positive integers, because there is a common difference between each
successive term. The common difference can also be called the first difference.
Linear functions can model any pattern where the first difference is the same
number.
2
𝑦 = 𝑥−1
3
1, 3, 5, 7, …
∨∨∨
+2 +2 +2
First Difference
Exponential Function: a function of the form 𝑓(𝑥) = 𝑎𝑏 𝑥 , where a and b are
constants, and 𝑎 ≠ 0, 𝑏 > 0 , and 𝑏 ≠ 1.
Exponential functions are most easily recognized by the variable in the
exponent. The values of 𝑓(𝑥) are either increasing (exponential growth) if 𝑎 >
0 and 𝑏 > 1 or decreasing (exponential decay) if 𝑎 > 0 and 0 < 𝑏 < 1 .
Exponential functions can model geometric sequences, where the domain
is the set of positive integers, because each successive term is multiplied by
the same number, called the common ratio. Exponential functions can
model any pattern where the next term is obtained by multiplying each
successive term by the same number.
Quadratic Function: a function that can be written in the
form 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑥 where 𝑎 ≠ 0.
Quadratic functions are most easily recognized by the 𝑥 2 term. The graph is a
𝑦 = 2𝑥 + 2
1, 3, 9, 27, …
∨∨∨
∙3 ∙3 ∙3
Common Ratio
parabola. A quadratic function can be formed by multiplying two linear functions.
The quadratic function to the right can also be written as 𝑓(𝑥) = (2𝑥 − 3)(𝑥 − 1).
𝑦 = 𝑥2 + 2
1, 4, 9, 16, …
To determine if a pattern or a sequence can be modeled by a quadratic function,
∨∨∨
you have to look at the first and second difference. The second difference is the
+3 +5 +7 (1st
difference between the numbers in the first difference. If the first difference is not Diff)
the same number but the second difference is, then the pattern or sequence can
∨∨
be modeled by a quadratic function.
+2 +2 (2nd
Diff)
Example 1:
Determine if the pattern 1, 3, 9, 19, … would be modeled by a linear function, an exponential function or a
quadratic function.
Example 2:
Determine if the pattern 2, 4, 8, 16, … would be modeled by a linear function, an exponential function or a
quadratic function.
Example 3:
Determine if the pattern would be modeled by a linear function, an exponential function
or a quadratic function. If possible, write an expression for each.
1.
2.
1
2
3
4
1
3.
1
5.
2
3
81, 27, 9, 3, …
2
4.
10, 18, 28, 40, …
6.
8, 16, 24, 32, …
4
3
Example 4:
Using a graphing calculator determine the quadratic function modeled by the given data.
𝑥
𝑓(𝑥)
1
1
2
9
3
23
4
43
5
69
6
101
Input the data into a TI-84 calculator list




Enter the following information into your lists by pushing STAT followed by
EDIT (#1).
If you have values in your lists already, you can clear the information by
highlighting the name of the list, then pushing CLEAR and ENTER. Do not
push DEL or it will delete the entire list.
Enter the x values into L1 and the 𝑓(𝑥) values into L2.
Push 2nd MODE to get back to the home screen.
Make a scatter plot






Push 2nd Y= to bring up the STAT PLOT menu.
Select Plot 1 by pushing enter or 1.
Turn Plot 1 on by pushing ENTER when ON is highlighted.
Make sure that the scatter plot option is highlighted. If it isn’t, select it by
pushing ENTER when the scatter plot graphic is highlighted.
The Xlist should say L1 and the Ylist should say L2. If it doesn’t, L1 can be
entered by pushing 2nd 1 and L2 by 2nd 2.
To view the graph, you can push GRAPH. If you want a nice viewing, first press
ZOOM, then arrow down to option 9 ZOOMSTAT and either push ENTER or
push 9.
Creating a quadratic regression equation




You do not have to graph a function to create a regression, but it is
recommended that you compare your regression to the data points to determine
visually if it is a good model or not.
From the home screen push STAT, arrow right to CALC and either push 5 for
QuadReg or arrow down to 5 and push ENTER. (To do an exponential
regression, push 0 for ExpReg or arrow down to 0 and push enter.)
Type 2nd 1 (the comma is located above the 7) 2nd 2, VARS arrow right to YVARS select FUNCTION and Y1 then press enter.
The quadratic regression is 𝑓(𝑥) = 3𝑥 2 − 𝑥 − 1. It has been pasted into Y1 so
that you can push GRAPH again and compare your regression to the data.
Example 5:
Find the regression equation. Round to the nearest thousandth if necessary.
a.
Given the table of values use a graphing calculator to find the quadratic function.
𝑥
𝑓(𝑥)
b.
0
-6
1
-21
2
-40
3
-57
4
-66
5
-61
Use a graphing calculator to find a quadratic model for the data.
𝑥
𝑓(𝑥)
1
3
2
1
3
1
4
3
5
7
6
13
c.
The cell phone subscribers of the small town of Herriman are shown below. Find an
exponential equation to model the data.
Year
Subscribers
Additional Notes/Examples:
1990
285
1995
802
2000
2,259
2005
6,360
2010
17,904
Unit 5 Lesson 8 –Building Functions that Model Relationships
Ready, Set, Go! Assignment - 5.8
Name__________________________________________
Date_______ Hour_____
Ready
Determine if the pattern would be modeled by a linear function, an exponential function or a quadratic
function.
1.
2.
3, 11, 19, 27, …
1, 3, 9, 27, …
3.
7, 10, 13, 16, 19,…
4.
5.
4, 7, 12, 19, 28, …
6.
Set
Find the regression equation. Round to the nearest thousandth if necessary.
7.
Use a graphing calculator to find an exponential model for the data.
𝑥
𝑓(𝑥)
8.
2
3.3
4
2.9
6
5.6
8
11.9
10
19.8
Use a graphing calculator to find a quadratic model for the data.
𝑥
𝑓(𝑥)
9.
0
1.1
1
1.67
5
2.59
9
4.37
13
6.12
17
5.48
21
3.12
Use a graphing calculator to find a quadratic model for the data.
𝑥
𝑓(𝑥)
-2
1.1
-1
3.3
1
2.9
1
5.6
2
11.9
3
19.8
Go!
Determine if the data is best modeled by an exponential or quadratic function. Then find the appropriate
equation. Round your answer to the nearest thousandth if necessary.
10.
The following table shows how many miles per gallon a car gets at different speeds.
Speed (mph)
Miles per gallon
11.
years.
45
25
50
28
55
30
60
29
65
25
The following table shows the amount of money an investor has in an account each year for 10
Year
Value of account
12.
time.
40
23
1996
5,000
1998
5,800
2000
6,800
2002
7,900
2004 2006
9,300 11,000
The value of a car depreciates over time. The table shows the value of a car over a period of
Year
Value ($)
0
18,500
1
15,910
2
3
4
13,682.60 11,767.04 10,119.65
13.
The table shows the average monthly number of flights made each year by a charter airline that
was founded in 2000.
Year
Flights
2000
17
2001
20
2002
24
2003
30
2004
33
2005 2006
30
24
2007
18
Unit 5 Lesson 9 –Graphing Quadratics by Plotting Points
Task 5.9
Name__________________________________________
Date_______ Hour_____
Review
1. Evaluate the function f(x) = 3x + 1 for x = 2, 0, –2
2. Evaluate the function g(x) = (x+3)2 – 1 for x = 2, 0, –2
Example #3
Graph f (x) = x2 by completing the table.
x
-3
-2
-1
0
1
2
3
f (x) = x2
Example #4
Graph 𝑓(𝑥) = (𝑥 − 2)2 + 3 by completing the table.
f (x) = (x - 2)2+3
x
-1
0
1
2
3
4
5
Example #5
Graph f (x) = x2 + 6x + 9 by completing the table.
x
-6
-5
-4
-3
-2
-1
0
f (x) = x2 + 6x + 9
Unit 5 Lesson 9 –Graphing Quadratics by Plotting Points
Assignment 5.9
Name__________________________________________
Date_______ Hour_____
1. Evaluate the function h(x) = x2 – x + 4 for x = 2, 0, –2
2. Evaluate the function k(x) = 3x2 – 5x + 1 for x = 2, 0, –2
Graph each equation without using a graphing calculator. Fill in the table of values.
3.
x
-3
-2
-1
0
1
2
3
𝒇(𝒙) = 𝒙 − 𝟐
4.
x
𝒇(𝒙) = 𝒙𝟐 − 𝟐
-3
-2
-1
0
1
2
3
5.
x
-2
-1
0
1
2
3
4
𝒇(𝒙) = (𝒙 − 𝟏)𝟐 − 𝟑
6.
x
𝒇(𝒙) = (𝒙 + 𝟏)𝟐 − 𝟐
-4
-3
-2
-1
0
1
2
7.
x
-2
-1
0
1
2
3
4
𝒇(𝒙) = −𝒙𝟐 + 𝟐𝒙 + 𝟏