Quadratic Functions Factoring and Solving

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-5Factoring – Difference of square.……………………..……12

Lesson 3A-6Factoring – Perfect Square Trinomial.…………………… 13

Lesson 3A-7Solving Quadratics – Square Roots………………………15

Lesson 3A-8Solving Quadratics – Factoring……………………………16

Lesson 3A-9 Solving Quadratics – Completing the Square……………18

Lesson 3A-10 Solving Quadratics – Quadratic Formula.……………… 20

This packet belongs to:

______

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:

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

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, , 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:

Find the ______of each term.

Identify the ______that are present in each number.

______the common factors together.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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)

3A-1 Practice

Identify whether or not the following are quadratic expressions. Justify your decision.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Find the greatest common factor of the given terms for each problem.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

1. 361842

5. 63

6. 19x23x

7. -15x45

8. -36x-81

9. -28x-40

10. -108240

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Application 3A-1

There is not an application for this lesson. Please begin preparing for your mastery check.

Lesson 3A – 2: Factoring – The GCF Method

Learning Target: I can factor the GCF from a polynomial.

A.SSE.3a

ExpressionFactors

4x2y4, x2(x and x), and y

(x-2)(x+3)(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 ______, greatestcommon factor, and ______that out before proceeding with other strategies.

GCF Method:

Example 1:

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Factor 2x + 6x2

______( )

______( + )

This is the “factored form”.

1. Find the ______of all terms.

2. Write down your ______, then a set of parenthesis.

3. To find out what goes in the parenthesis, you______!

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

FACTOR THESE EXAMPLES:

Ex. 2 12a2 – 18a Ex. 3 -15 + 25m

Ex. 4 4x3 – 24x2 + 12x Ex. 59x2y - 27xy2

Ex. 6 –3x2 – 7x – 9EX. 749 – 7x

3A-2 Practice

Factor the following quadratic expressions.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

5. 8x+4

10. 64x-8

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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 .

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:______

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!!!

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example 1

x2 + 6x + 8

(x )( x )

Example 2

x2 + 12x – 45

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example3

y2 – 10y + 16

Example 4

x2 – 3x – 28

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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 .

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:

Think about where our numbers come from when we multiply binomials.

Example 1 5x2 + 11x + 2Example 2 4x2 – 9x + 5

Example 3 2x2 + 4x – 6Example 4 6n2 – 11n – 10

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.

Example 5 -6x2 + 13x – 2

3A-4 Practice

Factor the following quadratic expressions.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Application 3A-4

There is not an application for this lesson. Please begin preparing for your mastery check.

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 ______.*

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example 1:

Example 2:

Example 3:

Example 4:

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

3A-5 Practice

Determine whether the binomial is a difference of squares. If so, factor. If not, explain why it is not.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Application 3A-5

There is not an application for this lesson. Please begin preparing for your mastery check.

Lesson 3A – 6: Factoring – Perfect-Square Trinomials

Learning Target: I can factor perfect-square trinomials.

A.SSE.2

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 ______.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example 1:

Example 2:

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example 3:

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

3A-6 Practice

Determine whether each trinomial is a perfect square. If so, factor. If not, explain.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

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.

A.REI.4b

When solving linear equations in Unit 2 we relied heavily on inverse operations. This will remain true when solving quadratics.

Recall from Unit 1that the inverse operation of squaring a number is taking the ______.

So , and

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 ______.

Steps for Solving Quadratic Equations Using Square Roots

Simplify each side of the equation by ______and ______.
Move all ______to one side of the equation.
Get by itself using ______.
Take the ______of both sides of the equation.
There will ALWAYS be a ______AND a ______.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

Example 2:

Example 4:

Example 3:

Example 5:

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)

3A-7 Practice

Solve each quadratic equation.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

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Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

Application 3A-7

Collect 3A-7Solving Quadratics Applicationsheet 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

Make sure each equation is equal to ______.
Check to see if there is a ______that can be factored out.
______.
Set each factor equal to ______.
______each equation.

Example 1:Example 2:

Solve: x2 + 3x – 18 = 0Solve: 4x2 + 20x – 24 = 0

Example 3:Example 4:

Solve: 5x2 + 10x = 0Solve: 4x2 + 12x + 9 = f(x)

3A-8 Practice

Find the zeros of each function by factoring.

1. hx x2 6x 9 2. f x 2x2 9x 4 3.

Solve each quadratic function by factoring.

4.9x2 4 = 12x 5. 16x2 9 6.

7. 8. 9.

10. 11.______3x2x = 0______12.

Application 3A-8

Collect 3A-8Solving Quadratics Applicationsheet 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.

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

Move the ______to the other side.
Find the number that “completes the square” using the formula,
______that number to ______sides!
______. NOTE: = #
Take the ______of each side.
Solve for x.

Example 1:Example 2:

Solve: Solve:

Example 3:Example 4:

Solve:Solve:

3A-9 Practice

Solve each equation.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

10.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

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

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 ______.

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:

Steps for Solving Quadratic Equations by Using the Quadratic Formula

Write the equation in ______.
Determine the value for ______.
Plug these numbers into the ______.
Simplify.

Example 1:Example 2:

Find the zeros:Solve:

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 ______.

Example 3:

Solve:

Example 4:

Solve:

There will be one real solution when:

There will be no real solutions when:

3A-10 Practice

Solve each equation using the Quadratic Formula.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

Find the zeros of each function using the Quadratic Formula.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1

Application 3A-10

Collect 3A-10Solving Quadratics Applicationsheet from the Algebra Embassy.

Unit 3A: Modeling and Analyzing Quadratic Functions (Factoring and Solving)1