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Unit 3A
Quadratic Functions – Factoring and Solving
Table of Contents
Title
Page #
Glossary.……..………………………………………………………….……… 2
Lesson 3A-1 Intro Quadratics and GCF…………….……………………….3
Lesson 3A-2 Factoring – The GCF Method..……………………….………. 5
Lesson 3A-3 Factoring – a = 1.………………………………………………. 6
Lesson 3A-4 Factoring – a ≠ 1.………………………………………………. 9
Lesson 3A-5 Factoring – Difference of square.……………………..……12
Lesson 3A-6 Factoring – Perfect Square Trinomial.…………………… 13
Lesson 3A-7 Solving Quadratics – Square Roots ………………………15
Lesson 3A-8 Solving Quadratics – Factoring……………………………16
Lesson 3A-9 Solving Quadratics – Completing the Square……………18
Lesson 3A-10 Solving Quadratics – Quadratic Formula.……………… 20
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Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
1
Glossary
Complete Factorization over the Integers:
Completing the square:
Difference of two squares:
Parabola:
Perfect square trinomial:
Quadratic equation:
Quadratic function:
Root:
Standard form of a quadratic function:
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
2
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Lesson 3A – 1: Intro to Quadratics and Greatest Common Factor
Learning Target: I can find the greatest common factor of a set of algebraic
terms.
A.SSE.3
Vocabulary:
- Quadratic equation
- Quadratic function
- Parabola
Guided Notes:
Which of the following are quadratic?
A) y = 3x + 2
B) y = -2x2 + x - 1
C) y = 3x4 - 3x2 + 2
The graph of the most basic quadratic function, 𝑓(𝑥) = 𝑥 2 ,
looks like this:
The first tool you will need in your mathematician’s tool belt is
finding the Greatest Common Factor (GCF) of a set of terms.
A __________ is anything that is being multiplied by
something else in math.
There are 3 simple steps to finding the GCF:
1. Find the _______________ of each term.
2. Identify the ___________ that are present in
each number.
3. ___________ the common factors together.
Example 1:
Find the GCF: 12v2 and 18v
Example 2:
Find the GCF: 2x2, 6x and 8
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
3
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3A-1 Practice
Identify whether or not the following are quadratic expressions. Justify your
decision.
1) 𝑦 = 4𝑥 2 − 7𝑥 + 3
3) 𝑦 = 2𝑥 + 7𝑥 − 4
2) 𝑦 = 7𝑥 2 − 3
4) 𝑦 = 3𝑥 2
5) 𝑦 = 4𝑥 4 − 3𝑥 2 + 9
Find the greatest common factor of the given terms for each problem.
6. −17𝑥 2
19x
23x
3𝑥
7. 35𝑥 2
-15x
45
3. 3𝑥 2
7𝑥
8. −21𝑥 2
-36x
-81
4. 32𝑥 2
8𝑥
9. −12𝑥 2
-28x
-40
5. 14𝑥 2
21𝑥
10. 42𝑥 2
-108𝑥 2
240𝑥 2
1. 36
18
2. 27𝑥 2
42
63
Application 3A-1
There is not an application for this lesson. Please begin preparing for your mastery
check.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
4
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Lesson 3A – 2: Factoring – The GCF Method
Learning Target: I can factor the GCF from a polynomial.
A.SSE.3a
Expression
4x2y
(x-2)(x+3)
Vocabulary:
- Complete
Factorization over
the Integers
Factors
4, x2 (x and x), and y
(x-2) and (x+3)
YOU TRY: What are the factors of:
-2x2
_______ and _________
x(x-1)(2x+3)
_________ and __________ and ___________
____________ is the process of breaking down algebraic expressions into the
most simplified form of all of its factors.
There are many strategies to factoring, and we will learn several of them in this
unit. No matter what strategy we use, we will ALWAYS first look for a _______,
greatest common factor, and __________ that out before proceeding with other
strategies.
GCF Method:
Example 1:
Factor 2x + 6x2
______ (
______ (
1. Find the _______ of all terms.
)
+
2. Write down your ________, then a
set of parenthesis.
)
This is the “factored form”.
3. To find out what goes in the
parenthesis, you ______________
_______________________!
FACTOR THESE EXAMPLES:
Ex. 2
12a2 – 18a
Ex. 3
-15 + 25m
Ex. 4
4x3 – 24x2 + 12x
Ex. 5
9x2y - 27xy2
Ex. 6
–3x2 – 7x – 9
EX. 7
49 – 7x
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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3A-2 Practice
Factor the following quadratic expressions.
1. 4𝑣 2 + 8𝑣
6. 9𝑥 3 − 3𝑥 + 12
2. 𝑥 2 − 3𝑥
7. −4𝑥 2 − 16𝑥 − 20
3. 2𝑏 2 − 18𝑏
8. −12𝑥 2 𝑦 − 18𝑥𝑦 − 28𝑥𝑦 2
4.6𝑏 2 + 12𝑏 2
9. 48𝑥 2 − 16𝑥 + 40
5. 8x+4
10. 64x-8
Application 3A-2
There is not an application for this lesson. Please begin preparing for your mastery
check.
Lesson 3A – 3: Factoring when a = 1
Learning Target: I can factor quadratic trinomials of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄.
A.SSE.3a
We now need to learn a strategy for factoring polynomials that have 3 terms,
________________. After looking for and factoring out a ________, we will see if
we can factor more using this method.
WARMUP EXERCISE:
1. Think of 2 numbers whose sum = 8 and whose product = 12: _________
2. Think of 2 numbers whose sum = 5 and whose product = -14:_________
3. Think of 2 numbers whose sum = 16 and whose product = 15:________
4. Think of 2 numbers whose sum = -14 and whose product = 40:________
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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5. MULTIPLY
a)
(x + 6)(x + 3)
b)
(x + 7)(x - 2)
What do you notice about the sum of the constants in each factor?
What do you notice about the product of the constants in each factor?
FACTORING A TRINOMIAL is kind of like multiplying binomials in reverse!!!
Example 1
x2 + 6x + 8
Factors of
(x
)
)( x
Example 3
y2 – 10y + 16
Factors of
Sum
Sum
Example 2
x2 + 12x – 45
Example 4
x2 – 3x – 28
Factors of
Sum
Factors of
Sum
Note: Not every trinomial is factorable. If you have tried every combination and
none of them work, write “non factorable” and move on!
3A-3 Practice
1. 𝑚2 + 16𝑚 + 60
3. 𝑎2 + 3𝑎 − 40
5. 𝑥 2 + 𝑥 − 90
2. 𝑚2 − 10𝑚 + 24
4. 𝑣 2 + 13𝑣 + 40
6. 𝑟 2 − 12𝑚 + 20
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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7. 𝑥 2 + 10𝑥 + 21
12. 𝑛2 + 12𝑛 + 27
17. 𝑝2 − 2𝑝 + 48
8. 𝑟 2 + 8𝑟 + 12
13. 𝑎2 + 3𝑎 − 4
18. 𝑛2 + 3𝑛 + 10
9. 𝑥 2 − 3𝑥 − 54
14. 𝑝2 − 2𝑝 + 48
19. 𝑚2 − 7𝑚 + 6
10. 𝑛2 − 14𝑛 + 48
15. 𝑛2 − 16𝑛 + 60
20. 𝑛2 − 17𝑛 + 70
11. 𝑟 2 − 13𝑟 + 42
16. 𝑥 2 − 6𝑥 − 7
21. 𝑛2 + 17𝑛 + 70
Application 3A-3
There is not an application for this lesson. Please begin preparing for your mastery
check.
Lesson 3A – 4: Factoring when a≠1
Learning Target: I can factor quadratic trinomials of the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄.
A.SSE.3a
If the leading coefficient is a number other than 1, we need to consider how this
will affect our first term in each parenthesis:
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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Think about where our numbers come from when we multiply binomials.
Example 1 5x2 + 11x + 2
Factors
of
Factors
of
Sum of outer and
inner products
Example 3 2x2 + 4x – 6
Factors
of
Factors
of
Example 2 4x2 – 9x + 5
Factors of
Factors of
Example 4
Sum of outer and
inner products
Factors of
Sum of outer and
inner products
6n2 – 11n – 10
Factors of
Sum of outer and
inner products
There is an additional consideration when factoring trinomials in which “a” is
negative.
When the leading coefficient is negative, ___________________ from each term
before using other factoring methods.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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Example 5 -6x2 + 13x – 2
Factors
of
Factors
of
Sum of outer and
inner products
3A-4 Practice
Factor the following quadratic expressions.
1. 3𝑥 2 − 5𝑥 − 28
6. 3𝑥 2 − 14𝑥 − 5
2. 7𝑥 2 + 60𝑥 − 27
7. 6𝑥 2 − 32𝑥 + 40
3. −5𝑥 2 − 16𝑥 − 12
8. 35𝑥 2 + 130𝑥 − 225
4. 6𝑥 2 + 𝑥 − 1
9. 3𝑥 2 + 17𝑥 + 24
5. 5𝑥 2 + 37𝑥 + 42
10. −7𝑥 2 + 27𝑥 − 18
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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11. 5𝑥 2 + 46𝑥 + 48
16. 3𝑥 2 − 28𝑥 − 20
12. 7𝑥 2 + 67𝑥 − 30
17. 27𝑥 2 − 240𝑥 + 192
13. 7𝑥 2 − 2𝑥 + 14
18. −10𝑥 2 + 63𝑥 + 49
Write your
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14. 9𝑥 2 + 9𝑥 − 28
15. 3𝑥 2 + 19𝑥 − 14
19. 27𝑥 2 + 9𝑥 − 60
20. 18𝑥 2 − 44𝑥 − 30
Application 3A-4
There is not an application for this lesson. Please begin
preparing for your mastery check.
Vocabulary:
- Difference of Two
Squares
Lesson 3A – 5: Factoring – Difference of Squares
Learning Target: I can factor the difference of two
squares.
A.SSE.2
Sometimes patterns can be used to help the factoring process along. The next
couple of lessons are going to focus on some special cases of trinomials.
The difference of 2 squares has the form _________________.
A polynomial is a difference of squares if:
 There are ____ terms, one ____________ from the other.
 Both terms are _____________________.*
* A term is a perfect
square when it is the
product of a number
and itself.
Examples: 4, x2, 9y2
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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Example 1:
8𝑥 2 − 18
Example 3:
𝑥 6 − 7𝑦 2
Example 2:
9𝑝4 − 16𝑞 2
Example 4:
1 − 25𝑘 8
3A-5 Practice
Determine whether the binomial is a difference of squares. If so, factor. If
not, explain why it is not.
1. 1 − 4𝑥 2
6. 81𝑥 2 − 1
2. 2𝑝8 − 18𝑞 6
7. 4𝑥 4 − 9𝑦 2
3. 16𝑥 2 − 4𝑦 5
8. 𝑥 8 − 50
4. 𝑥 2 − 81
9. 𝑥 6 − 9
5. 𝑠 2 − 42
10. 2𝑚2 − 8𝑟 6
Application 3A-5
There is not an application for this lesson. Please begin preparing for your mastery check.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
12
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Lesson 3A – 6: Factoring – Perfect-Square Trinomials
Learning Target: I can factor perfect-square trinomials.
A.SSE.2
Vocabulary:
- Perfect Square
Trinomials
Another special case for factoring is when you are given a
perfect-square trinomial.
A trinomial is a perfect square if:
 The ________ and ________terms are ________________.
 The ____________ term is ______________ one factor from the
_____________ and one factor from the _____________.
Example 1:
𝑥 2 + 12𝑥 + 36
Factors of
Example 3:
8𝑥 2 − 40𝑥 + 50
Factors of
Sum
Sum
Example 2:
4𝑥 2 − 24𝑥 + 9
Factors of
Sum
Notice the
pattern!
3A-6 Practice
Determine whether each trinomial is a perfect square. If so, factor. If not,
explain.
1. 4𝑥 2 − 12𝑥 + 9
3. 𝑥 2 + 4𝑥 + 4
2. 𝑥 2 + 9𝑥 + 16
4. 4𝑥 2 − 14𝑥 + 49
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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5. 9𝑥 2 − 6𝑥 + 4
8. 2𝑥 2 − 8𝑥 + 8
6. 9𝑥 2 − 12𝑥 + 4
9. 𝑥 2 − 6𝑥 + 9
7. 𝑥 2 − 4𝑥 − 4
10. 𝑥 2 − 6𝑥 − 9
Application 3A-6
There is not an application for this lesson. Please begin preparing for your mastery
check.
Lesson 3A – 7: Solving Quadratics – Square Roots
Learning Target: I can solve simple quadratic equations by taking square
roots.
Vocabulary:
A.REI.4b
- Root
When solving linear equations in Unit 2 we relied heavily on
inverse operations. This will remain true when solving quadratics.
Recall from Unit 1 that the inverse operation of squaring a number is taking the
_____________________.
So √𝑥 2 = 𝑥, √4 = 2 and √4𝑥 2 = 2𝑥
We will now use this fact to solve
simple quadratic equations.
Example 1 x2 = 49
Not every quadratic equation will be a one step solution. At times, we have to work
to get the x2 ______________ before taking the ___________________.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
14
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Steps for Solving Quadratic Equations Using Square Roots
1. Simplify each side of the equation by __________ and ___________________.
2. Move all ____________ to one side of the equation.
2
3. Get x by itself using ___________________.
4. Take the __________________ of both sides of the equation.
5. There will ALWAYS be a ________________ AND a __________________.
Example 3: 2(𝑥 2 − 5) = −𝑥 2 − 1
Example 2: 2𝑥 2 − 338 = 0
1
Example 5: (𝑥 + 6)2 = 49
Example 4: 2 𝑥 2 + 3 = 12
3A-7 Practice
Solve each quadratic equation.
4. (𝑥 + 4)2 = 121
7. (𝑥 + 3)2 + 6 = 18
2. 3𝑥 2 − 7 = 47
5. (2𝑥 − 3)2 = 9
8. (2𝑥 + 6)2 − 8 = 24
3. 𝑥 2 + 11 = 16
6. (𝑥 − 7)2 = 99
9. 𝑥 2 − 21 = 5
1. 𝑥 2 + 4 = 29
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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10. 3(𝑥 + 4)2 = 9
11. 3(𝑥 2 − 4) = 2𝑥 2 − 1
2
12. 5 𝑥 2 − 3 = 7
Application 3A-7
Collect 3A-7 Solving Quadratics Application sheet from the Algebra Embassy.
Lesson 3A – 8: Solving Quadratics – Factoring
Learning Target: I can solve quadratic equations by factoring.
A.REI.4b
Isolating the x2 term is great when that is the only x term in the equation. Unfortunately
for us, most quadratic functions are not that simple. For this reason, we need to look at
several more tools for solving quadratic equations.
This method requires use of the factoring skills we mastered in lessons 2 through 6.
Steps for Solving Quadratic Equations Using Factoring
1. Make sure each equation is equal to ________.
2. Check to see if there is a _______ that can be factored out.
3. _________.
4. Set each factor equal to ________.
5. ________ each equation.
Example 1:
Solve: x2 + 3x – 18 = 0
Example 2:
Solve: 4x2 + 20x – 24 = 0
Example 3:
Solve: 5x2 + 10x = 0
Example 4:
Solve: 4x2 + 12x + 9 = f(x)
3A-8 Practice
Find the zeros of each function by factoring.
1.
x2
x
2.
x2
x
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
3. 𝑔(𝑥) = 𝑥 2 + 𝑥 − 20
16
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Solve each quadratic function by factoring.
4. 9x2 4 = 12x
5. 16x2 9
6. 𝒚𝟐 + 𝟕𝒚 + 𝟏𝟐 = 𝟎
7.𝑥 2 − 7𝑥 − 44 = 0
9. 𝑧 2 − 12𝑧 + 27 = 0
10. 4𝑘 2 − 35𝑘 − 9 = 0
8. 𝑥 2 − 12𝑥 = −36
11.3x2
x=0
12. −2𝑥 2 − 5𝑥 + 10 = −2
Application 3A-8
Collect 3A-8 Solving Quadratics Application sheet from the Algebra Embassy.
Lesson 3A – 9: Solving Quadratics – Completing the Square
Learning Target: I can solve quadratic equations by completing the square.
A.REI.4a & A.REI.4b
Many quadratic equations contain expressions that are not _______
__________ AND cannot be easily ____________. These
equations require us to learn yet another method of solving.
Vocabulary:
- Complete the
Square
In this lesson, we are going to be using algebraic properties to rewrite any quadratic
expression as a perfect square.
Steps for Solving Quadratic Equations by Completing the Square
1.
2.
3.
Move the ___________ to the other side.
Find the number that “completes the square” using the formula,
______ that number to ______ sides!
4.
5.
6.
_________. NOTE: (𝑥 + 2) = #
Take the ____________ of each side.
Solve for x.
𝑏 2
Just cut the middle
number in half and
square it!
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
17
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Example 1:
Solve: 𝑥 2 = 6𝑥 + 4
Example 3:
Solve: 4𝑥 2 + 32𝑥 + 16 = 0
Example 2:
Solve: 𝑥 2 − 2𝑥 − 1 = 0
Example 4:
Solve: 𝑥 2 + 4𝑥 − 4 = 0
3A-9 Practice
Solve each equation.
1. 𝑥 2 + 8𝑥 − 20 = 0
3. 3𝑥 2 + 12𝑥 − 12 = 0
2. 𝑥 2 + 6𝑥 − 18 = 0
4. 4𝑥 2 − 8𝑥 − 4 = 0
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
18
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5. 𝑥 2 − 8𝑥 − 36 = 0
8. 2𝑥 2 + 8𝑥 − 15 = 0
6. 𝑥 2 + 6𝑥 = 16
9. 𝑥 2 + 2𝑥 − 3 = 0
7. 2𝑥 2 = 3𝑥 + 4
10. 𝑥 2 = 6𝑥 + 4
Application 3A-9
There is not an application for this lesson. Please begin preparing for your
mastery check.
Lesson 3A – 10: Solving Quadratics – The Quadratic Formula
Learning Target: I can solve quadratic equations using the quadratic
formula.
A.REI.4b
Vocabulary:
- Standard Form of a
Quadratic Equation
We have learned several methods for
solving quadratic equations now. Now its
time to show you one last method before
moving on. This final method can be used on
________ quadratic equation as long as it is
written in ________ _______.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
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When we say standard form of a quadratic, we mean it looks like this:
By completing the square on the standard form of a quadratic equation, you can
derive the Quadratic Formula.
The quadratic formula, in case you can’t read the unicorn’s handwriting (or Mr.
Thomaswick’s), is:
This IS on the
formula sheet!
Steps for Solving Quadratic Equations by Using the Quadratic Formula
1. Write the equation in _______________.
2. Determine the value for _________________.
3. Plug these numbers into the ____________________.
4. Simplify.
Example 1:
Example 2:
Find the zeros: 𝑥 2 + 9𝑥 − 14 = 0
Solve: −𝑥 2 + 3𝑥 = −4
Although we have come to expect two solutions for every quadratic formula, there
are scenarios where this is not the case. It is possible for a quadratic equation to
have only _____________ or even _____________.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
20
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Example 3:
Solve: 2𝑥 2 − 12𝑥 = −18
Example 4:
Solve: 𝑥 2 − 6𝑥 + 11 = 0
There will be one real solution when:
There will be no real solutions when:
3A-10 Practice
Solve each equation using the Quadratic Formula.
3. 2𝑥 2 − 16𝑥 = −32
2
1. 2𝑥 + 5 = 2𝑥
2. 2𝑥 2 − 3𝑥 = 8
4. 3𝑥 2 + 3 = 10𝑥
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
21
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5. 𝑥 2 = −6𝑥
8. 𝑥 2 − 5 = 𝑥
6. 𝑥(𝑥 − 3) = 4
9. −3𝑥 2 + 4𝑥 = 4
7. 7𝑥 2 − 3 = 0
10. 4𝑥 2 − 28𝑥 = −49
Find the zeros of each function using the Quadratic Formula.
11. 𝑓(𝑥) = 𝑥 2 + 7𝑥 + 10
13. ℎ(𝑥) = 3𝑥 2 − 5𝑥
12. 𝑔(𝑥) = 3𝑥 2 − 4𝑥 − 1
14. 𝑟(𝑥) = 𝑥 2 + 6𝑥 + 12
Application 3A-10
Collect 3A-10 Solving Quadratics Application sheet from the Algebra Embassy.
Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)
22