topic 24: factoring and quadratic equations factoring and

Topic 24: Factoring and quadratic equations 433

Copyright © 2015 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc.

FACTORING AND QUADRATIC EQUATIONS Lesson 24.1 Rectangles and factors

24.1 OPENER

Write the length and width for each algebra tile rectangle.

1. 2.

3. 4.

24.1 CORE ACTIVITY

Use algebra tiles to represent each polynomial and find its factors. For each polynomial, sketch and/or describe the rectangle you made, and state the factors. 1. x2 + 3x 2. x2 + 5x + 6

434 Unit 8 – Quadratic functions and equations


Work with your partner to build each polynomial with tiles. Then sketch or describe the rectangle, and state the factors.

3. x2 + 6x + 5 4. x2 + 7x + 10 5. x2 + 7x + 6 6. x2 + 8x + 15 7. x2 + 6x + 9 8. x2 – 4 For each algebra tile rectangle, write an expression for its area by counting the tiles. Then complete the area model by referring to the tile rectangle, and use it to identify the factors. 9.

Expression:

Factors:

10.

Expression:

Factors:

11. What is the first thing you should do when factoring a polynomial? Discuss your answer with your partner, and be prepared to share your response with the class.



12. Complete the math journal entry.

Describe in words and with examples how to find the factors of a number.

What does it mean to factor a polynomial?

24.1 CONSOLIDATION ACTIVITY For each quadratic expression, work with your partner to:

• Draw an area model to represent the polynomial, using algebra tiles to build a rectangle if you like; • Write the factored form of the quadratic expression; and • Multiply the factors to check your answer.

Quadratic Expression Area Model Factored Form and Check

Example:

x2 + 4x + 3

x 3

x x2 3x

1 x 3

Factors: (x + 1) (x + 3)

Check by multiplying:

(x + 1) (x + 3)

(x + 1)(x) + (x + 1)(3)

x · x + 1 · x + x · 3 + 1 · 3

x2 + x + 3x + 3

x2 + 4x + 3

1. x2 + 6x + 8

Factors:


2. x2 + 7x + 10

Factors:




Quadratic Expression Area Model Factored Form and Check

3. x2 + 18x + 77

Factors:


4. x2 + 2x – 8

Factors:


5. x2 + 8x + 16

Factors:


6. x2 – 9

Factors:




HOMEWORK 24.1 Notes or additional instructions based on whole-class discussion of homework assignment:

1. Write expressions for the area, length, and width of each of these algebra tile rectangles.

a.

Area =

Length =

Width =

b.

Area = Length = Width =

2. Write expressions for the total area, the length, and the width of each area model. Hint: Look for common factors in the terms in each row and each column.

Exam

ple x2 3x

2x 6

Total = x2 + 3x + 2x + 6

= x2 + 5x + 6

Length = x + 2

Width = x + 3

a. x2 5x

3x 15

Total =

Length =

Width =

b. x2 11x

6x 66

Total =

Length =

Width =

c. x2 -4x

2x -8

Total =

Length =

Width =



3. Use the area model to find the factors of each polynomial. Draw a tile rectangle if that helps you complete the area model. (The guidelines can help you separate the tiles.)

Example: x2 + 5x + 6

Draw Tiles Area Model

Factors of x2 + 5x + 6: (x + 2) and (x + 3)

x2 2x

3x 6

a. x2 + 7x + 10 Draw Tiles Area Model

Factors of x2 + 7x + 10:

b. x2 + 6x + 8 Draw Tiles Area Model

Factors of x2 + 6x + 8:



4. What is the product of 2 and 7?

What is the sum of 2 and 7?

5. Multiply the factors (x + 2) and (x + 7) using a method of your choice.

6. How are your answers to questions 4 and 5 connected?

7. Homer and Bart are factoring the trinomial x2 + 11x + 24. Homer says that the trinomial factors as (x + 8) (x + 3), while Bart says that it factors as (x + 3) (x + 8). Explain why Homer and Bart are both correct.

8. Could Homer and Bart have had a similar disagreement if they had been factoring x2 + 6x + 9? Explain why or why not. (Hint: What was special about the rectangle you made for x2 + 6x + 9 during the lesson?)



STAYING SHARP 24.1 Pr

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1. Graph the points in the table on the grid provided.

x y

-3 1.5

-1 -2.5

2 -1

4 5

2. Graph the function rule y = -2x + 5 on the grid provided.

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3. Write a function rule for the linear function represented by the graph.

4. A sandwich shop prices its party subs at $17.50 per foot, with a $23 assembly charge. Write a function rule describing how the price of a party sub (p) depends on the sub’s length in feet (l).

What is the price of a party sub that is 8 feet long?

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5. Describe a situation in which there is a functional dependence, and identify the independent and dependent variables in this situation. (Make up your own context, or use a context from a problem you encountered this year.)

Independent Dependent

Cause Effect

Before After

Input Output

What you do What happens

6. Rewrite this expression as simply as possible:

3(4x + 5) + 2(x – 7) + 6x



Lesson 24.2 Reversing area models to factor

24.2 OPENER

In the example square box problem at the right, notice that the product of 8 and 2 is 16, and also that the sum of 8 and 2 is 10.

Now, find the pairs of numbers that will work for the four square box problems shown below.

1. 2. 3. 4.

After completing the square box problems, discuss the following questions with your partner, and be ready to share your ideas with the class:

• In what ways do square box problems connect with factoring quadratic expressions? • Do you think there is always a way to complete a square box problem? Why or why not? • Do you think there is ever more than one way to complete a square box problem? Why or why not?

24.2 CORE ACTIVITY

1. Examine the examples, then answer the following questions about each part of the area model for quadratic expressions of the form x2 + bx + c.

Part 1 a. Which term of the polynomial goes

here?

Part 2

Part 3 Part 4 b. Which term of the polynomial goes

here?



2. How are the contents of Parts 2 and 3 of the area model related to the contents of Parts 1 and 4?

3. How are the contents of Parts 2 and 3 related to the middle term, bx, in the polynomial x2 + bx + c?

4. Factor x2 + 11x + 18 using the area model.

5. In your own words, describe a perfect square trinomial.

Then, give another example of a perfect square trinomial, and identify its factors.

6. Explain why the polynomial x2 – 4 is called a difference of two squares. Then, write another example of a difference of two squares, and identify its factors.

7. Factor the quadratic expression x2 + 5x + 8. (Follow the example of question 4 if you are having trouble.)



24.2 CONSOLIDATION ACTIVITY You have investigated several different methods for factoring quadratic expressions. In this activity, you and your partner will match quadratic expressions with their factors.

Example:

x2 + 2x – 3 = (x + 3) (x – 1)

Hint: Some factors may be used multiple times, and other factors may not be used at all!

When you and your partner have found the factors for all of the quadratic expressions, record them in the spaces below.

1. x2 + 6x =

2. x2 + 8x + 12 =

3. x2 + 4x – 12 =

4. x2 – 5x – 14 =

5. x2 – 15x + 56 =

6. x2 – 64 =

Of the four polynomials below, two are not factorable (using integer coefficients). Identify the non-factorable polynomials and explain or justify why they cannot be factored.

Polynomial Factorable or not? Explanation or justification

7. x2 + 6x + 10

8. x2 – 16x + 55

9. x2 + 14x + 13

10. x2 – 12x + 18




Use the area model or another method to factor each trinomial. Example: x2 + 5x + 6

Factor pairs of 6x2:

1x and 6x 2x and 3x -1x and -6x -2x and -3x

Factor pair with a sum of 5x: 2x and 3x

Area Model

x2 3x

2x 6

What are the factors of x2 + 5x + 6? By looking at the common factors of the terms in each row and column, we can see the factors: (x + 3) and (x + 2)

1. x2 + 10x + 21 Factor pairs of 21x2:

Factor pair with a sum of 10x:

Area Model

What are the factors of x2 + 10x + 21?

2. x2 + 14x + 40

Factor pairs of 40x2:


Area Model


3. x2 + 14x + 33

Factor pairs of _____:

Factor pair with a sum of _____:

Area Model


4. x2 + 13x + 30



Area Model


5. x2 + 13x + 42

Area Model


6. x2 + 9x + 20




Because the product of two positive numbers is never negative, you will sometimes need to consider factor pairs that may have different signs. For example: –14 = –1 · 14 = –2 · 7 = –7 · 2 = –14 · 1.

7. x2 + 3x – 28 Factor pairs of -28x2:


Area Model

What are the factors of x2 + 3x – 28?

8. x2 + 7x – 30



Area Model

What are the factors of x2 + 7x – 30?

9. x2 – 5x – 24

Area Model


10. x2 – 13x + 40

What are the factors of x2 – 13x + 40? (Hint: The sum of two positive factors isn’t -13.)




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1. A hardware store rents a floor sander for $12 an hour, plus a check-out fee of $20. Amar models this pricing plan with the function y = 12x + 20. a. What do the variables x and y represent in Amar’s

model? b. What are the domain and range of the function

that Amar wrote?

2. In the context of the problem situation in Question 1, what are the domain and range?

Explain why your answer to this question differs from your answer to Question 1b.

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3. Is the function represented by this table linear or nonlinear? Justify your answer.

x y

1 1

2 3

3 6

4 10

5 15

4. Does the rule y = 2x (x + 1) represent a linear or a nonlinear function? Explain.

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5. A total of $360 is divided into equal shares. If Bob receives four shares, Carol receives three shares, and Ted receives the remaining two shares, how much money did Carol receive?

6. Deborah is enlarging a photo to fit on a base that is 16 inches wide. If the original photo is 3 inches wide and 5 inches tall, how tall will the enlarged photo be?



Lesson 24.3 Solving quadratic equations by completing the square

24.3 OPENER

1. Simplify the following expressions.

a.

b.

c.

d.

2. What do you notice? How are the answers related to the numbers you squared?

24.3 CORE ACTIVITY 1. Solve the following quadratic equations using what you know about square roots.

a. b.

c. d.

2. Identify which quadratic expressions are perfect squares.

a. 4x2 b. x2 + 4x + 4

c. x2 + 1 d. x2 + 6x + 4

e. (x2 + 1)2 f. x2 + 6x + 9

g. 4x2 + 4x h. x2 + 10x + 25

3. Solve the equation x2 + 10x + 25 = 9.

4. Can you use the same technique you used in question 3 to solve the equation x2 + 6x + 11 = 5? Explain why or why not.



5. Solve the equation x2 + 6x + 11 = 5 by completing the square. Use algebra tiles to build a concrete model.

6. Solve the equation x2 + 8x = 6 by completing the square.

7. Use an area model to help you solve x2 + 5x = 6 by completing the square. How will you make your square if you only have five x-tiles?



8. Complete the square on the general quadratic equation ax2 + bx + c = 0 to derive the quadratic formula. Write the correct steps along with the new equation for each step.

Given equation ax2 – bx + c = 0

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

Step 6.

Step 7.

Step 8.



24.3 CONSOLIDATION ACTIVITY 1. Use algebra tiles to complete the square on the equation x2 - 8x + 4 = -2. Then solve the equation.

2. Write the equation represented by the area model. Then, solve the equation.



3. Cassandra is trying to solve the quadratic equation x2 + 10x – 4 = 6. Her steps are shown here. To check her answers, Cassandra graphs both expressions in the original equation and compares the x-values of the intersection of the graphs to the solutions she calculated by completing the square. She can see that they do not match. Where did Cassandra go wrong?

Cassandra’s steps: Cassandra’s graph:




1. Solve the following equations. If one side of the equation is not a perfect square trinomial, use the method of completing

the square to solve. Draw a sketch of your diagram if you used one.

a. x2 + 12x + 36 = 25 b. x2 + 4x + 4 = 10 c. x2 – 10x + 4 = -12 d. x2 + 6x + 12 = 5

e. x2 – 8x + 10 = 2 f. x2 + 7x + 3 = 1

2. Maria is trying to solve the equation x2 +8x + 4 = -2 by completing the square. She is using algebra tiles to help her visualize the process. She doesn’t think she arrived at the correct answer, so she is trying to describe her process to a friend. Where did she go wrong?

“First, I modeled the left-hand side using 1 x2-tile, 8 x-tiles, and 4 unit-tiles and I modeled the right hand side using 2 negative unit-tiles. Then, I formed part of a square using the x2-tile and the 8 x-tiles. I saw that I needed 16 unit-tiles to complete the square, and since I had 4 already, I added 12 more unit-tiles to the left hand side to complete the square. So

now my equation is (x + 4)2 = -2, which simplifies to x + 4 = . I can’t take the square root of a negative number, so this would tell me there are no solutions. But the equation should have solutions because I graphed it and it crossed the x-axis. Help me!”




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1. A skater is moving at a constant rate. This table shows his distance along a track at various times. Find the skater’s rate and complete the table.

Time (minutes)

Distance (meters)

4 500

7

11

20 2900

2. This graph depicts the progress of two cyclists. If the vertical axis is in miles, find the rate of the faster cyclist in miles per hour.

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3. This chart compares two runners.

Runner Distance (miles)

Time (hours)

Bill 11 2

Ted 16 3

Based on the information in this chart, state which runner has the faster rate. Justify your answer.

4. Marty rides his bicycle from home to work at an average rate of 12 miles per hour. If it takes him 20 minutes to get to work, how many miles is his home from his work?

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5. Tom drove 290 miles from his college to home, and his trip used 23.2 gallons of gasoline. His sister Andrea drove 225 miles from her college to home, and she used 15 gallons of gasoline.

How would you figure out whose vehicle had better gas mileage? Explain.

6. A machine can bind 500 yearbooks in 3 hours. How many hours will it take to bind 1800 yearbooks?



Lesson 24.4 More about completing the square

24.4 OPENER Drake wants to analyze one of his soccer kicks. Here are the data from Drake's kick:

What is the vertex of the graph? What does this point represent?

24.4 CORE ACTIVITY

1. What is another way to write the symbolic form of the quadratic function y = x2 + 6x + 11?

2. Graph the function y = (x + 3)2 + 2 and sketch the graph.

3. What is the vertex of this graph? What is the equation of the axis of symmetry?

4. Where do you see the vertex and axis of symmetry in the transformed function rule? Explain.

5. How is this function transformed from the parent function?



6. Complete the table to transform the given quadratic functions.

Function Sketch of concrete representation

Function rule in vertex form

Vertex Axis of symmetry Description of transformation of

the parent function

a. y = x2 + 6x + 7

b. y = x2 + 5x + 7

7. Consider the function y = 2(x – 5)2 – 3.

a. What are the vertex and the axis of symmetry of the parabola that represents this function?

b. Use the function rule to find another point on the graph. Graph the function using the vertex, axis of symmetry, and the additional point you found.



24.4 CONSOLIDATION ACTIVITY 1. Complete the square to convert the general form of y = x2 + 4x + 9 into vertex form.

a. What is the vertex form of the function rule? Sketch your concrete model to support your answer.

b. What is the vertex and axis of symmetry of the graph?

C. Find another point on the graph and sketch a graph of the function.

2. Follow the steps to complete the square on y = 3x2 + 18x + 21.

Step 1: Use the Distributive Property to write y = 3x2 + 18x + 21 as the product of a constant and a quadratic with a leading coefficient of 1.

Step 2: Complete the square on the quadratic expression you wrote in step 1. Use algebra tiles if necessary.

Step 3: Apply the Distributive Property to the result.



3. What is the vertex and the axis of symmetry of the graph of y = 3x2 + 18x + 21? Sketch the graph by hand and describe the

transformations on the parent function.

4. Follow the same process as in question 3 to complete the square on y = 5x2 – 20x – 5.

a. What is the vertex form of the function?

b. What is the vertex and axis of symmetry of the parabola that represents the function?

c. Sketch the graph of this function.




1. Given the following functions, state the vertex and what transformations on the parent function are needed to make the graph of the given function.

a. y = (x – 3)2 + 5

b. y = (x + 1)2 – 4

c. y = ½(x + 2)2 + 3 2. Transform the function y = x2 – 4x + 1 into vertex form using a diagram. Sketch a model of your square. Then state the

vertex and axis of symmetry of the graph of the function.

3. Use the process of completing the square to write y = 2x2 +12x + 14 in vertex form.

4. Now, state the vertex of the graph that represents the function in question 3. What are the transformations of this function

from the parent function? Sketch a graph of this new function.



5. Two friends, Ryan and Tressa, are on vacation. They decide to go bungee jumping off of a platform. Their height from the ground can be modeled by the function y = 4x2 – 32x + 80, where x represents the time after they jumped and y represents their height above the ground.

a. How high is the platform off the ground?

b. Write the function rule in vertex form.

c. What is the vertex of the parabola that represents this function? What does this point mean in the problem?

d. If Ryan and Tressa are 32 feet above the ground after 2 seconds, when else are they 32 feet above the ground? Explain your reasoning.

e. At what time do Ryan and Tressa reach the platform again? Explain how you know.

f. Sketch a graph of the parabola that represents this situation. Be sure to correctly label your axes.



STAYING SHARP 24.4

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1. What is the slope of the line that passes through the points (2,5) and (7,3)?

2. What is the y-intercept of a line with slope 3 that passes through the point (4,1)? (Use of the grid is optional.)

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3. What is the slope of line ℓ?

4. Raúl is examining how the thickness of the trunks of oak trees depends on their age. He models the growth with the function d = 0.8t + 4, where d represents the diameter of the trunk in centimeters and t represents the age of the tree in years.

What does the value 0.8 mean in this context?

What does the value 4 mean in this context?

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5. Write an equation of the line that passes through the point (4,-6) and has a slope of -3.

6. Write an equation that represents the line that passes through the points (5,4) and (-5,0).



Lesson 24.5 Solving x2 + bx + c = 0 by factoring

24.5 OPENER 1. Find four pairs of values for a and b that result in a

product of 6.

a b a · b

6

6

6

6

2. Find four pairs of values for a and b that result in a product of 0.

a b a · b

0

0

0

0

3. What must be true about the factors a and/or b for their product to be zero?

24.5 CORE ACTIVITY

Part 1. Solving, factoring, and the zero product property Find the values of x that satisfy the following equations.

1. x (x – 4) = 0 2. (x – 7) (x – 11) = 0

3. (x + 5) (x + 2) = 0

4. (x + 2) (x – 2) = 0

5. (x – 3) (x – 3) = 0

6. (2x – 4) (x – 1) = 0

7. Based on questions 1–6, how do you think factoring can be useful to solve quadratic equations?

8. Use what you have learned in recent lessons to solve the equation x2 – 5x + 6 = 0.

9. Are x = 3 and x = 8 both solutions of the equation x2 – 11x + 24 = 6? Discuss this question with your partner, and be prepared to explain your answer and your reasoning to the class.

10. Solve the equation x2 – 11x + 24 = 6 by following these steps:

Given equation x2 – 11x + 24 = 6

Step 1. Use properties of equality to make one side of the equation equal to 0

Step 2. Factor the quadratic expression

Step 3. Use the zero product property to write two simpler equations

Step 4. Solve each simpler equation



11. Solve the equation x2 – 14x + 48 = 15.

12. Examine the steps of your solution to question 11. Describe each step to make a list you can use to solve similar problems.

Part 2. The Caldas Garden Problem

Your neighbors, Mr. and Mrs. Caldas have a small rectangular yard that measures 171 square feet. They have decided to create a square garden in one corner of their yard and to pour concrete on the rest of the yard to make a patio. The Caldas' plans show that they want the patio to be 15 feet wide on one side of the garden and 5 feet wide on the other side. Mr. and Mrs. Caldas want to find the dimensions of their square garden.

13. Let x represent the length of the side of the square garden.

Write an equation to represent the relationship between the area of the backyard and the side of the square.

14. Find the solutions of your equation using factoring.

15. Do both solutions of the equation lead to feasible side lengths for the square garden? Explain your answer.



24.5 CONSOLIDATION ACTIVITY Use the zero product property to find the solutions of each equation.

1. (x – 4)(x – 2) = 0

2. (x – 13) (x – 17) = 0

3. (x + 5) (x – 9) = 0

Solve each equation by factoring and using the zero product property.

4. x2 – 7x + 10 = 0

5. x2 + 10x + 21 = 0

6. x2 + 3x – 28 = 0

7. Marco first solves the equation (x + 2)(x – 3) = 0. Show the steps needed to find the values of x that satisfy this equation.

8. Marco then solves the equation (x + 2)(x – 3) = 6. His work is shown below. Next to each step in his solution process, write a mathematical justification for what Marco did.

9. Why did Marco have to use more steps to solve the equation in question 8 than the equation in question 7?



10. Below is Marisa’s work when asked to solve the equation (x + 6)(x – 2) = 9. Are her solutions valid?

Marisa’s Steps Reasons

Find values of x so that (x + 6)(x – 2) = 9 Given problem

x + 6 = 9 x – 2 = 9 Set each factor to equal 9

x + 6 + – 6 = 9 +– 6 x – 2 + 2 = 9 + 2 Making use of the addition property of equality

x = 3 and x = 11 Combining like terms to find the values of x.

In this course, you have practiced the Mathematical Problem-solving Routine, which asks you to look back on your solution. One strategy to check your solution is to substitute the values x = 3 and x = 11 back into the original equation, (x + 6)(x – 2) = 9. Use this looking back strategy to show Marisa whether her solutions are valid or not.

Solve each equation for x, showing all of the steps needed.

11. x2 + 4x – 12 = 0

12. x2 – 5x + 11 = 35

13. (x + 6)(x – 2) = 9




1. How many different factor pairs can you find for 12… a. if both factors in the pair are positive?

b. if the factors can be positive, negative, or zero?

c. Are factors ever repeated across pairs?

2. How many different factor pairs can you find for 0?

3. What is true of any factor pair for 0?

4. Use the zero product property to find the solutions of each equation. a. (x – 2) (x – 6) = 0

b. (x) (x – 5) = 0

c. (x + 4) (x – 3) = 0

d. (x + 7) (x + 1) = 0

5. Factor the trinomial by any method, then use the zero product property to find the solutions of each equation. a. x2 – 13x + 40 = 0

b. x2 + 3x – 40 = 0

c. x2 + 13x + 40 = 0

d. x2 – 8x + 16 = 0

e. x2 – 25 = 0

6. Danny is trying to solve the equation x2 + 6x + 5 = 12. He writes the following steps: x2 + 6x + 5 = 12

(x + 1) (x + 5) = 12

x = -1 or x = -5

Alisa thinks something is wrong. She says, “I don’t think those solutions work.”

a. Evaluate the initial equation for x = -1 and x = -5. Are these values solutions of the equation? (Do they make the equation true?)

Alisa then advises, “I think the problem is that one side of the equation isn’t zero.”

b. Use your algebraic skills to rewrite Danny’s equation in a way that will let you apply the zero product property. Then find the correct solutions of the initial equation.




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1. Yuri wants to learn to play the guitar. He found a used guitar for $55 and lessons for $12 each. He has saved $147 from his part-time job. Write an inequality to represent the number of lessons Yuri can take with the money he has saved.

2. What value of p makes this equation true?

6(p – 2) = 36 – 10p

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3. Graph the solutions of the inequality 3x + 1 > 7 on this number line.

4. Solve the following inequality:

5x – 4 > 3x + 6

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5. The formula for potential energy is P = mgh, where P is potential energy, m is mass, g is gravity, and h is height. Write an expression to represent g in terms of the other variables. (Solve for g.)

6. Kofi is trying to figure out how many buses his 150 students will need for a college visit. Each bus holds 48 people, so he writes 150 = 48b.

Solve this equation.

Is your solution the answer to Kofi’s question? Explain, and give the answer to Kofi’s question if it is different from your solution.



Lesson 24.6 More about solving by factoring

24.6 OPENER In the general form for a quadratic polynomial, ax2 + bx + c, let a = 1, b = 14, and c = 24.

1. After substituting values for a, b, and c, write your polynomial.

2. In the area models below, Parts 1 and 4 have been provided. Find Parts 2 and 3, then circle the model that represents the polynomial you wrote in question 1.

x 1 x 2

x x2 x x2

24 24 12 24

x 3 x 4

x x2 x x2

8 24 6 24

3. Why are there exactly four area models being examined to help you determine the factors of x2 + 14x + 24?

24.6 CORE ACTIVITY

The function that models the height of a water balloon as a function of time, f(t) = -16t2 + 64t, can be used to find how long it will take a water balloon to hit the ground after it is launched.

Because the height of the balloon will be 0 feet when it hits the ground, solving the equation 0 = -16t2 + 64t provides the answer to this question.

1. What are you being asked to find?

2. Rewrite the polynomial -16t2 + 64t as a product.



3. Continue factoring and use the zero product property to find the solutions of 0 = -16t2 + 64t. Hint: You can use the simpler polynomial you wrote in question 2 to make this easier.

4. Refer back to the context of the problem and your answer to question 1. Explain what each solution means in the context of the problem, and determine which solution corresponds to what you are being asked to find.

5. Solve the equation 0 = -16t2 + 64t again using the quadratic formula.

6. What factorizations of the polynomial 2x2 + 2x – 4 does the animation illustrate?

7. What does it tell you about a polynomial if you can represent it with a tile rectangle in more than one way?

8. Solve the equation 2x2 + 2x – 4 = 36 by factoring.



24.6 CONSOLIDATION ACTIVITY Work with a partner to answer the following questions.

1. A rectangle has an unknown width and a length of 3 inches more than the width. a. Provide a sketch of the rectangle with the width and length (in terms of the width) labeled.

b. Write a function that could be used to represent the area of the rectangle, a, in terms of the width of the rectangle, w.

c. Write an equation that you could solve to determine the width of the rectangle if the area of the rectangle was 28

square inches.

d. What would the length of the rectangle be if the area of the rectangle was 28 square inches? Explain how you know.

e. Find the dimensions of the rectangle if the area of the rectangle was 70 square inches.

2. The height of a diver above the water during a dive can be modeled by the function h(t) = -16t2 – 8t + 48, where h is the height of the diver above the water and t is time in seconds. a. Write an equation that you could solve to determine how long it will take the diver to hit the water.

b. Solve the equation from part (a) by factoring.

c. Write an equation that you could solve to determine at what time the diver is 40 feet above the water.

d. Solve the equation from part (c) by factoring.

e. Graph the function h(t) on your calculator.

f. What do you notice about the graph and your answers to (b) and (d)?




Find the solutions of each equation using factoring, graphing, the quadratic formula, or another method. (You may find graph paper helpful.)

For each problem, state the method you decided to use and explain why; then show all the work for your chosen method.

1. x2 – 11x + 28 = 0 2. x2 – 10x + 16 = 0

3. x2 + 8x – 33 = 0 4. x2 – 13x + 54 = 14

5. x2 – 15x – 28 = 6 6. 2x2 – 16x + 30 = 0

7. 3x2 + 15x + 150 = 0 8. 2x2 – 24x + 92 = 22




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1. Nina has a collection of stamps and coins. She has 561 items in all. The number of stamps is twice the number of coins. Identify the variables and write a system of linear equations to model this situation.

2. Use tables to find the solution of this system of equations:

y = 2x + 5

y = 3x – 1

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3. Is the ordered pair (2,-1) a solution of this system of equations? Provide evidence for your answer.

3x + 2y = 4

-2x + 2y = 24

4. Find the solution of this system of equations:

2x + 3y = 8

x = 4y – 7

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5. Find the solution of this system of equations:

4x + 7y = 5

2x – 3y = 9

6. Solve this system by substitution.

y = 2x + 5

y = 2x – 3

What does your solution mean?



Lesson 24.7 Connecting solution methods

24.7 OPENER

Roan tried to solve the equation x2 + x = -2 by factoring. Here is his work:

Because there are no factors of 2x2 that add to be x, he realizes the expression on the left cannot be factored. He concludes that the equation has no solution.

Is Roan’s conclusion correct? Explain your thinking by using another solution method, such as graphing, completing the square, _or the quadratic formula.

24.7 CORE ACTIVITY The Martinez family has a large lot. They have decided to create a square garden in one corner of the lot. They want to put a walkway around the garden. The walkway will be 5 feet wide on one side of the square and 8 feet wide on the other side. The total area of the garden and the walkway must be 700 square feet.

1. Write the dimensions of the combined garden and walkway. Length = ____________ Width = ____________

2. If A represents the area of the combined garden and walkway in square feet, write a function rule for the area in terms of the side of the square, x.

A = ________________

3. Use the information you have about the area of the garden and walkway to write an equation.

4. If necessary, rewrite the equation you created for question 3 so that one side is equal to 0.

5. Solve the equation by factoring.

6. Each solution makes the equation true, but do all of the solutions make sense in the context of the problem situation? Explain.



7. Solve the equation by graphing.

8. Solve the equation using the quadratic formula.

9. Solve the equation by completing the square.



10. Complete the table to connect solution methods.

Factors of the polynomial

x2 + 13x – 660

Solutions of the equation

x2 + 13x – 660 = 0 by factoring

x-intercepts of the graph of

y = x2 + 13x – 660

Solutions of the equation x2 + 13x – 660 = 0

using the quadratic formula

Solutions of the equation x2 + 13x – 660 = 0

using completing the square

11. Use a graph to solve the equation x2 + 2x + 7 = 0. Sketch your graph on these axes.

12. Apply your knowledge of the connections among solution methods to consider what the results of Question 11 mean. Discuss your ideas with your partner, and be prepared to explain your answer and your reasoning to the class.

ONLINE ASSESSMENT Today you will take an online assessment.




1. Find the solutions of each equation using factoring.

a. x2 – 13x + 36 = 0

b. x2 + 5x – 9 = 27

c. 2x2 + 6x – 8 = 0

2. Consider the equation x2 + 4x – 21.

a. Find the solutions of the equation x2 + 4x – 21 = 0 using the quadratic formula.

b. Find the solutions of the equation x2 + 4x – 21 = 0 by factoring.

c. Find the solutions of the equation x2 + 4x – 21 = 0 by completing the square.

d. Use this graph of the function y = x2 + 4x – 21 to find the solutions of the equation x2 + 4x – 21 = 0.



e. Complete this table to connect the four methods you have learned to solve a quadratic equation.

Factors of the polynomial x2 + 4x – 21

Solutions of x2 + 4x – 21 = 0

by factoring

Solutions of x2 + 4x – 21 = 0 using the quadratic formula

Solutions of x2 + 4x – 21 = 0 using

completing the square

x-intercepts of the graph of

y = x2 + 4x – 21

f. How are the solutions of the equation x2 + 4x – 21 = 0 connected to the graph of the function y = x2 + 4x – 21?

g. How are the factors of the polynomial x2 + 4x – 21 connected to the solutions of the equation x2 + 4x – 21 = 0?

3. The length of a rectangular window is 5 feet more than its width, x. The window’s area is 36 square feet.

a. Write an equation to find the dimensions of the window. b. Solve your equation for x. c. Find the width and length of the window. d. -9 is also a solution of the equation you wrote.

Explain why this solution does not work as an answer to the question.



4. A biologist is studying a pond. She models the population of ducks, y, as a function of the density of algae in the pond, x, with the function y = -20x2 + 140x + 1000. If there are 840 ducks (y = 840), what is the density of algae in the pond?

a. How can you use the given population, 840, to write an equation from the function rule? b. Rewrite the equation so that one side is 0. c. Find a common factor shared by all the terms in your equation. d. Solve this simpler equation using factoring, graphing, or the quadratic formula. e. Which of the two solutions is feasible in the context of the problem?

5. The marketing team for Capital Computer Company is trying to find the most profitable selling price for the company's new laptop computer. After much research, the team has decided that the function N = -100p2 + 300,000p represents the expected relationship between N, the net sales in dollars, and p, the retail price of the laptop computer.

a. Graph the function. What does the graph tell you about the relationship between the price and the net sales? What are the x-intercepts, and what do they mean in the context of the scenario?

b. Solve the equation -100p2 + 300,000p = 0 by factoring. What are your solutions?

c. How do the x-intercepts on the graph compare with the solutions you found by factoring?’

d. By looking at your graph, how could you determine what price corresponds to the highest net sales in dollars?




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1. Use the Laws of Exponents to rewrite this expression with only two exponents.

y7 · (x3)4 · (y2)5 · x -6

2. Write the number 0.0000082 in scientific notation.

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3. There are approximately 6 × 1023 molecules of gas in each liter of air. If a classroom contains approximately 2 × 105 liters of air, approximately how many molecules of gas does it contain?

4. This table represents a function.

x 0 1 2 3 4

y 3 5 11 21 35

Explain why this function is not linear.

Using the table, determine whether the function is quadratic or exponential.

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5. Taeko models a population of bacteria with the function n = 45 · 1.07t. Its graph is below.

What is the value of the y-intercept of the graph?

What does the y-intercept mean in this situation?

6. In words, describe how the graph of y = 2x2 – 3 will compare to the graph of the function y = x2.

Then, sketch a graph of y = 2x2 – 3 here.



Lesson 24.8 Connecting functions to graphs

24.8 OPENER

Solve 2x2 – 14x = - 20 using two different methods. Explain one advantage and one disadvantage of using each method you selected.

Circle method used Show solution Advantage Disadvantage

Graphing

Quadratic formula

Completing the square

Factoring

Graphing

Quadratic formula

Completing the square

Factoring

24.8 CORE ACTIVITY

1. The graph of y = x2 + 4x + 4 is shown. What does this graph tell you about the solutions to the quadratic equation x2 + 4x + 4 = 0?



2. Consider the equation x2 — 5x — 36 = 0. What do the solutions to the equation tell you about the graph of y = x2 — 5x — 36?

3. Consider the graph shown.

a. What do you know about the function graphed here?

(4, 0)(1, 0)

b. What is the simplest equation that would have a graph like the one shown?

(4, 0)(1, 0)



4. What is the simplest equation that would have a graph like the one shown?

(4, 0)(1, 0)

5. Write the equation represented by this graph.



6. Do you think a quadratic function can have three or more zeros? Complete the statements to give both an algebraic and a graphical response to this question. Use the answer choices provided.



REVIEW ONLINE ASSESSMENT

Today you will review the online assessment.

Problems we did well on: Skills and/or concepts that are addressed in these problems:

Problems we did not do well on: Skills and/or concepts that are addressed in these problems:

Addressing areas of incomplete understanding

Use this page and notebook paper to take notes and re-work particular online assessment problems that your class identifies.

Problem #_____ Work for problem:






Next class period, you will take an end-of-unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts by reviewing related course materials.

Important skills and ideas you have learned in the unit Quadratic equations:

• Solving quadratic equations by graphing

• Solving quadratic equations with the quadratic formula

• Solving quadratic equations by completing the square

• Transforming quadratic functions using completing the square.

• Multiplying polynomials using algebra tiles, the area model, or other distributive property organizers

• Adding and subtracting polynomials

• Factoring using algebra tiles, the area model, or other distributive property organizers

• Solving quadratic equations by factoring

• Connecting solution methods for quadratic equations and choosing the appropriate tool for a problem

Homework Assignment

Part I: Study for the end-of-unit assessment by reviewing the key ideas in the topic as listed above.

Part II: Take the More practice from the topic Factoring and quadratic equations through the online services. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study.

Part III: Complete Staying Sharp 24.8




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1. The graph of the function y = -2x2 + 4x + 30 is shown. What is/are the solution(s) to the equation 0 = -2x2 + 4x + 30?

2. What type of function is represented by the data in the table? How do you know? If you can, write the function rule that represents these data. x -5 -4 -3 -2 -1

y 96 48 24 12 6

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5. How many solutions does the equation -3 = x2 – 6x + 6 have? What are they?

6. Factor the expression x2 – 2 – 8.



Lesson 24.9 Checking for understanding

24.9 OPENER

1. Look at the equation (x + 8) (x – 2) = 24. Which two values of x make this statement true? Show your work or describe how you found the two values.

2. Find someone in the class who solved the problem using a different method. Show their work in the box below and have them sign their name next to their work.

END-OF-UNIT ASSESSMENT Today you will take the end-of-unit assessment.

24.9 CONSOLIDATION ACTIVITY Think back to the first day of this class. 1. How are you different now than you were then?

2. How have your beliefs changed?

3. How have you changed as a mathematics learner?

4. Which three of the big ideas presented in the class posters have you found most valuable? Why?




Part I. Complete Staying Sharp 24.9.

Part II. This activity consists of a series of connected problems related to a situation called “Growing Staircases.” You will begin working on the problems now, and may continue working on them in the future as directed by your teacher. Each level of problems in this activity builds on the previous level. The more you think through the earlier levels, and the clearer your explanations of your answers, the more information you’ll have for later levels, and the further you will be able to progress.

As you work on these problems, think about the big ideas you have studied in this course—including ideas about mathematics, communication, persistence, problem solving, and growing your intelligence. Are you approaching these problems with more tools or with different strategies than you might have at the beginning of the year? If so, these changes are evidence of what you have learned in this course!

Level A

This is a staircase that goes up three steps.

1. How many blocks are needed for the first step?

2. How many blocks are needed for the second step?

3. How many blocks are needed for the third step?

4. How many blocks in all are needed to make this staircase of three steps? Explain how you know.

Level B

5. Draw additional blocks in the diagram to make the fourth step.

6. How many blocks in all are needed to make a

staircase with five steps? Explain your answer.

7. How many blocks does it take to build just the twelfth step? Explain your answer.

8. How many blocks in all are needed to make a staircase of ten steps? Explain your answer.

9. If a staircase has 105 blocks, how many steps does it have? Explain your answer.



Level C

10. How many blocks are needed to make just the 100th step? Explain how you know.

11. Write a rule to find the number of blocks needed for the xth step. Explain your rule.

12. Write a rule to find the total number of blocks needed to make a staircase with x number of steps.

Explain your rule.

13. Write a rule to find the number of steps in a staircase that has y total blocks. Explain your rule.

Level D

This set of staircases grows at a different rate.

1 step 2 steps 3 steps

14. How many blocks in all are needed to make a staircase with five steps? Explain your answer.

15. How many blocks make up the top step of a staircase with n steps? Explain your answer.

16. How many blocks make up the first level (the base) of a staircase with n steps? Explain your answer.

17. Given a staircase with 30 steps, explain a process that you could follow to determine the number of blocks necessary to build the staircase.



Level E

1 step 2 steps 3 steps

18. Using the pattern shown, find a general formula for determining the number of blocks needed to build a staircase with n steps.

19. Justify why your formula works.

20. For what numbers of steps will the staircase require an odd number of blocks to build? Explain and justify your answer.




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1. A right triangle has the dimensions shown.

a. What is the area of the triangle?

b. What is the perimeter?

2. A rectangle has an area of x2 + 8x + 15 square units. What are the lengths of the sides?

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3. Complete the square to solve the equation x2 + 8x + 1 = 0.

4. Refer to the triangle in question 1. The area of the triangle is 24 square units. What is the value of x and the length of each side? Explain how you arrived at your answer.

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5. What is the vertex of the graph that represents the function y = x2 + 4x – 5?

6. You know that the graph of a certain quadratic function crosses the x-axis at x = 4 and x = -2. What are two factors of the quadratic function that has this graph?

topic 24: factoring and quadratic equations factoring and

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