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**Solving Quadratic Equations by Factoring**

Chapter 10 Lesson 10-5

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Introduction This stAIR is designed for Ms. Goghar’s Grade 9 Algebra 1 class. You are expected to follow directions on each slide and navigate through this powerpoint carefully. Your goal is to understand the concept of solving quadratic equations and its relation to the roots (zeros) of quadratic functions and to learn how to solve quadratic equations by factoring.

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**To master the skill of solving quadratic equations by factoring**

Objective: To master the skill of solving quadratic equations by factoring

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**Pencil, Paper, and eraser TI-84 (optional)**

Required Materials: Pencil, Paper, and eraser TI-84 (optional)

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**The next 6 slides are Warm-Up Review Problems**

The next 6 slides are Warm-Up Review Problems. Being able to successfully solve these problems will ensure that you are ready to work on this lesson.

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**Solve this equation for n: What does n equal?**

Lesson Warm-Up #1 Solve this equation for n: What does n equal? -1 2

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You Are Right!!

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**Try Again!! Study this Example:**

First, you should subtract 6 from both sides. Now, what should the next step be? What does n equal? -1 2

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**Try Again!! First, you should subtract 6 from both sides.**

Now you need to divide both sides by 4. What does n equal? -1 2

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**Try Again!! First, you should subtract 6 from both sides**

Then, divide both sides by 4 What does n equal? -1 2

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**Solve for a: What does a equal?**

Lesson Warm-Up #2 Solve for a: What does a equal? 104 -40

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You Are Right!!! Solve for a:

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**Try Again!! Study this Example:**

NOW: Solve for a: Step 1: Add 9 on both sides So, Think: What is the 2nd Step you need to do to solve for a? What does a equal? 104 -40

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**Try Again! Solve for a: What does a equal? 104 -40**

Step 1: Add 9 on both sides Step 2: Multiply 8 to both sides What does a equal? 104 -40

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**Factor the expression completely: What is the Complete Factored form?**

Lesson Warm-Up #3 Factor the expression completely: What is the Complete Factored form?

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**Factor the expression:**

You Are Right!!! Factor the expression:

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**Try Again!!! Factor the expression completely:**

Step 1: Factor out the GCF. In this case, GCF = 2. Step 2: You need to factor Note: Factoring Difference of Perfect Squares: What is the Complete Factored form?

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**Try Again!!! Factor the expression completely:**

Step 1: Factor out the GCF. In this case, GCF = 2. Step 2: Factor So now, what is the Complete Factored form?

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**Factor the expression: Is it: OR**

Lesson Warm-Up #4 Factor the expression: Is it: OR

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**You Are Right!!! Factor the expression:**

This expression cannot be factored. Therefore, it is

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**Back to Lesson Warm-Up #4**

Try Again!!! Notes: Steps to Factor Polynomials: I Un-DISTRIBUTE GCF II Un-FOIL a2 - b2 III Un-FOIL ax2 + bx + c IV Polynomials that cannot be factored are PRIME Hint: Is it possible to factor the given expression? Does it follow any of the factoring rules listed above? Back to Lesson Warm-Up #4

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**Lesson Warm-Up #5 Factor the expression completely:**

What is the expression in factored form?

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**Factor the expression completely: Expression in Factored Form:**

You Are Right!!! Factor the expression completely: Expression in Factored Form:

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**Try Again!! Study the Notes Below: Steps to Factor Polynomials:**

I Un-DISTRIBUTE GCF II Un-FOIL a2 - b2 III Un-FOIL ax2 + bx + c IV Polynomials that cannot be factored are PRIME Factor completely:

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**Try Again!! Factor the expression completely: Step 1: (2c+___)(c+___)**

You need to think of factors of 14 that will work. Possibilities: 1 and 14; 2 and 7 Remember to check for the middle term when you FOIL. What is the expression in factored form?

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**Lesson Warm-Up #6 Factor the expression completely:**

What is the expression in factored form?

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**Factor the expression:**

You Are Right!!! Factor the expression:

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**Try Again!!! Study the Notes Below: Steps to Factor Polynomials:**

I Un-DISTRIBUTE GCF II Un-FOIL a2 - b2 III Un-FOIL ax2 + bx + c IV Polynomials that cannot be factored are PRIME Factor completely:

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**Try Again!!! Factor the expression completely: Step 1: (3p+___)(p+___)**

Think: What Factors of 20 will give you the correct middle term? What is the expression in factored form?

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Watch this video to get a sense of what we are going to do for the rest of this lesson: You need to return and continue working on this project after watching the video.

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**Zero-Product Property**

For every real number a and b, if ab = 0, then a = 0 or b = 0. Example: If , then x + 2 = 0 or x + 3 = 0

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**Checking Your Understanding #1:**

If x = 0, then You have 5 seconds to think before the answer appears…

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**Checking Your Understanding #2:**

If y = 0, then You have 5 seconds to think before the answer appears…

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**What should you do on the “Example” pages:**

Study the Example Question carefully and think about how you may want to solve it The worked out solution(s) and the final answer of the example will appear after a few seconds Study every step of the solution and the answer carefully so you will be able to solve similar problems later on When you are ready to continue, navigate to the next slide.

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**What should you do on the “You Try” pages:**

Study the “You Try” Question carefully and think about how you may want to solve it Solve the problem on the paper that you have with you The worked out solution(s) and the final answer of the problem will appear after seconds Check your work (step-by-step) with the worked out solution and final answer If you made mistakes in solving the problem on your paper, you should correct them When you are ready to continue, navigate to the next slide.

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**Using the Zero-Product Property, Solve:**

Example 1: Using the Zero-Product Property, Solve:

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Example 1 continues: Check your Solutions:

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**Using the Zero-Product Property, Solve:**

You Try #1: Using the Zero-Product Property, Solve:

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**Using the Zero-Product Property, Solve:**

You Try #2: Using the Zero-Product Property, Solve:

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**Using the Zero-Product Property, Solve:**

You Try #3: Using the Zero-Product Property, Solve:

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Quick Check Solve: A. B. C.

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Yippee! YOU GOT IT! Solve:

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**Oops! You need to review the Example and You Try Problems…**

Back to Examples

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Example 2: Solve by Factoring:

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You Try: Solve by Factoring:

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Quick Check Solve: A. B. C.

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Yippee! YOU GOT IT! Solve:

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**Oops! You need to review the Example and You Try Problems…**

Back to Examples

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Example 3: Solve by Factoring:

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You Try: Solve by Factoring:

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Quick Check Solve: A. B. C.

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Yippee! YOU GOT IT! Solve:

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**Oops! You need to review the Example and You Try Problems…**

Back to Examples

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**So, Width of material = x + 3 + 3 = x + 6 **

Example 4: The diagram below shows a pattern of an open-top box. The total area of the sheet is 288 inches square. The height of the box is 3 in. Therefore, 3-in. by 3-in. squares are cut from each corner. Find the dimensions of the box. Let x = width of a side of the box So, Width of material = x = x + 6 Length of material = x Length = x + 8

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**Length x Width = Area of Sheet**

Example 4 continued: The diagram below shows a pattern of an open-top box. The total area of the sheet is 288 inches square. The height of the box is 3 in. Therefore, 3-in. by 3-in. squares are cut from each corner. Find the dimensions of the box. Length x Width = Area of Sheet

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**Length x Width = Area of Sheet**

You try: Suppose that a box has a base with a width of x, a length of x + 3, and a height of 1 inch. It is cut from a rectangular sheet of material with an area of 130 inches square. Find the dimensions of the box. Length x Width = Area of Sheet

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**NOTES: What is the connection between solving quadratic equations by graphing and by factoring?**

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**Check Your Understanding #1:**

If the roots of the quadratic function g(x) are -2 and 2, what are the solutions of the equation: g(x) = 0

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**Check Your Understanding #2:**

If the solution of the quadratic equation h(x) are -3 and -5, what are the zeros of the function: y = h(x)

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Practice Problems #1: Solve:

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Practice Problems #2: Solve by Factoring:

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Practice Problems #3: Solve by Factoring:

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Quiz Time 1. Solve: A. B. C.

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**Oops!! Please review the Example below and Try Again!**

Using the Zero-Product Property, Solve: Back to QUIZ

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YES! YOU ARE RIGHT! NOW TRY QUESTION 2

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Quiz Time 2. Solve by Factoring: A. B. C.

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**Oops!! Please review the Example below and Try Again!**

Solve by Factoring: Back to QUIZ

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YES! YOU ARE RIGHT! NOW TRY QUESTION 3

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Quiz Time 3. Solve by Factoring: A. B. C.

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**Oops!! Please review the Example below and Try Again!**

Solve by Factoring: Back to QUIZ

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YES! YOU ARE RIGHT! NOW TRY QUESTION 4

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Quiz Time 4. Solve by Factoring: A. B. C.

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**Oops!! Please review the Example below and Try Again!**

Solve: Back to QUIZ

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**YES! YOU ARE RIGHT! You Have Done a Great Job on this StAIR!**

Please click Home to return to the First Slide to allow other students to work on this.

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