to ﬁnd common and x rewrite using the distributive ... · of signs of ac and b. l1 l2 learning...

Name_____________________________________ Class____________________________ Date________________

Lesson ObjectivesFinding common and binomial factorsof quadratic expressionsFactoring special quadratic expressions2

1

NAEP 2005 Strand: AlgebraTopic: Variables, Expressions, and Operations

Local Standards: ____________________________________

Vocabulary and Key Concepts.

Factoring Perfect Square Trinomials

a2 ! 2ab ! b2 " (a b)2

a2 # 2ab ! b2 " (a b)2

Factoring a Difference of Two Squaresa2 # b2 " (a b)(a b)

Factoring is

The greatest common factor (GCF) of an expression is

A perfect square trinomial is

The difference of two squares is

Examples.

Factoring When ac > 0 and b < 0 Factor x2 # 14x ! 33.

Step 1 Find factors with product ac and sum b.Since ac " 33 and b " #14, find negative factors with product 33 and sum #14.

Factors of 33 , ,

#14Sum of factors #34

1

Lesson 5-4 Factoring Quadratic Expressions

95Daily Notetaking Guide Algebra 2 Lesson 5-4

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Lesson 5-4 Factoring Quadratic Expressions 259

Factoring QuadraticExpressions

is rewriting an expression as the product of its factors. Theis a common factor of the

terms of the expression. It’s the common factor with the greatest coefficient and thegreatest exponent. You can factor any expression that has a GCF not equal to 1.

Finding Common Factors

Factor each expression.

a. 4x2 + 20x - 12

4x2 + 20x - 12 = 4x2 + 4(5x) - 4(3) Factor out the GCF, 4.

= 4(x2 + 5x - 3) Rewrite using the Distributive Property.

b. 9n2 - 24n

9n2 - 24n = 3n(3n) - 3n(8) Factor out the GCF, 3n.

= 3n(3n - 8) Rewrite using the Distributive Property.


a. 9x2 + 3x - 18 b. 7p2 + 21 c. 4w2 + 2w

11Quick Check

EXAMPLEEXAMPLE11

greatest common factor (GCF) of an expressionFactoring

5-45-4

Lessons 1-2 and 5-1

Simplify each expression.1. x2 + x + 4x - 1 2. 6x2 - 4(3)x + 2x - 3 3. 4x2 - 2(5- x)- 3x

Multiply.4. 2x(5 - x) 5. (2x - 7)(2x - 7) 6. (4x + 3)(4x - 3)

New Vocabulary • factoring • greatest common factor (GCF) of an expression• perfect square trinomial • difference of two squares

What You’ll Learn• To find common and

binomial factors ofquadratic expressions

• To factor special quadraticexpressions

. . . And WhyTo model the cross section ofa pipe, as in Example 8

11 Finding Common and Binomial Factors

Activity: Factoring

1. Since 6 ? 3 = 18, 6 and 3 make up a factor pair for 18. a. Find the other factor pairs for 18, including negative integers.b. Find the sum of the integers in each factor pair for 18.

2. a. Does 12 have a factor pair with a sum of -8? A sum of -9?b. Using all the factor pairs of 12, how many sums are possible?c. How many sums are possible for the factor pairs of -12?

x2 ± 5x – 1 6x2 – 10x – 3

–2x2 ± 10x

3(3x2 ± x – 6) 7(p2 ± 3) 2w(2w ± 1)

–19, –11, –9, 19, 11, 9yes; no

66

16x2 – 9

1a. –1 and –18, –2and –9, –3 and–6, 1 and 18, 2and 9

4x2 – 28x ± 49

4x2 – x – 10

Check Skills You’ll Need GO for Help

One number is a factor ofanother if the first dividesinto the second with noremainder.

Vocabulary Tip

5-45-4

259

1. PlanObjectives1 To find common and binomial

factors of quadraticexpressions

2 To factor special quadraticexpressions

Examples1 Finding Common Factors2 Factoring When ac ! 0 and

b ! 03 Factoring When ac ! 0 and

b " 04 Factoring When ac " 05 Factoring When a # 1 and

ac ! 06 Factoring When a # 1 and

ac " 07 Factoring a Perfect Square

Trinomial8 Real-World Connection

Math Background

A quadratic function of the formƒ(x) = ax2 + bx + c factors intotwo linear factors. When trying tofactor a quadratic function, welook for two factors whose sum isb and whose product is ac.

More Math Background: p. 236C

Lesson Planning andResources

See p. 236E for a list of theresources that support this lesson.

Bell Ringer Practice

Check Skills You’ll NeedFor intervention, direct students to:

Algebraic ExpressionsLesson 1-2: Example 4Extra Skills and Word

Problems Practice, Ch. 1

Modeling Data With Quadratic FunctionsLesson 5-1: Example 1Extra Skills and Word


PowerPoint

Special NeedsPair students who have difficulty remembering theirmultiplication facts with students who know themwell. Require students to check their answers bymultiplying their factors using FOIL.

Below LevelHave students make a table summarizing theprocedures in the examples for various combinationsof signs of ac and b.

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a. 4x2 + 20x - 12



b. 9n2 - 24n




a. 9x2 + 3x - 18 b. 7p2 + 21 c. 4w2 + 2w

11Quick Check

EXAMPLEEXAMPLE11


5-45-4

Lessons 1-2 and 5-1


Multiply.4. 2x(5 - x) 5. (2x - 7)(2x - 7) 6. (4x + 3)(4x - 3)







Activity: Factoring



x2 ± 5x – 1 6x2 – 10x – 3

–2x2 ± 10x

3(3x2 ± x – 6) 7(p2 ± 3) 2w(2w ± 1)

–19, –11, –9, 19, 11, 9yes; no

66

16x2 – 9


4x2 – 28x ± 49

4x2 – x – 10



Vocabulary Tip

5-45-4

259










Math Background











PowerPoint



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a. 4x2 + 20x - 12



b. 9n2 - 24n




a. 9x2 + 3x - 18 b. 7p2 + 21 c. 4w2 + 2w

11Quick Check

EXAMPLEEXAMPLE11


5-45-4

Lessons 1-2 and 5-1


Multiply.4. 2x(5 - x) 5. (2x - 7)(2x - 7) 6. (4x + 3)(4x - 3)







Activity: Factoring



x2 ± 5x – 1 6x2 – 10x – 3

–2x2 ± 10x

3(3x2 ± x – 6) 7(p2 ± 3) 2w(2w ± 1)

–19, –11, –9, 19, 11, 9yes; no

66

16x2 – 9


4x2 – 28x ± 49

4x2 – x – 10



Vocabulary Tip

5-45-4

259










Math Background











PowerPoint



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A quadratic trinomial is an expression in the form ax2 + bx + c. You can factormany quadratic trinomials into two binomial factors. First find two factors with aproduct ac and a sum b. Then find common factors.

If ac and b are positive, then the factors of ac are both positive.

Factoring When ac S 0 and b S 0

Factor x2 + 8x + 7.

Step 1 Find factors with product ac and sum b.

Since ac = 7 and b = 8, find positive factors with product 7 and sum 8.

Step 2 Rewrite the term bx using the factors you found. Group the remainingterms and find the common factors for each group. After removingcommon factors from each group, you should find two identical binomials.

x2 + 8x + 7

x2 + x + 7x + 7 Rewrite bx: 8x ≠ x ± 7x.(')'* (')'*x(x + 1) + 7(x + 1) Find common factors.

Step 3 Rewrite the expression as the product of two binomials.

x(x + 1) + 7(x + 1)

(x + 1)(x + 7) Rewrite using the Distributive Property.

Check (x + 1)(x + 7) = x2 + 7x + x + 7

= x2 + 8x + 7 ✓

Factor each expression. Check your answers.a. x2 + 6x + 8 b. x2 + 12x + 32 c. x2 + 14x + 40

If ac is positive and b is negative, then the factors of ac are both negative.

Factoring When ac S 0 and b R 0

Factor x2 - 17x + 72.


Since ac = 72 and b = -17, find negative factors with product 72 and sum -17.

Step 2 Rewrite the term bx using the factors you found. Then find commonfactors and rewrite the expression as the product of two binomials.

x2 - 17x + 72

x2 - 8x - 9x + 72 Rewrite bx.(')'* (')'*x(x - 8) - 9(x - 8) Find common factors.

(x - 9)(x - 8) Rewrite using the Distributive Property.

Factors of 72Sum of factors

!1, !72!73

!2, !36!38

!3, !24!27

!4, !18 !22

!6, !12!18

!8, !9!17

EXAMPLEEXAMPLE33

22Quick Check


1, 78

These are the only positive factors of 7.

EXAMPLEEXAMPLE22

260 Chapter 5 Quadratic Equations and Functions

You can use algebra tiles to factor the expression inExample 2.

(x ± 2)(x ± 4) (x ± 4)(x ± 8) (x ± 4)(x ± 10)

x2 x x x x x x x

x 1 1 1 1 1 1 1

A monomial is anexpression with one term. A binomial has two terms, and atrinomial has three terms.

Vocabulary Tip

260

2. Teach

Guided Instruction

ActivityIn order to factor quadraticexpressions, students must beable to find factor pairs forintegers. Have students completethe Activity to practice findingfactor pairs. Tell students it isimportant to include negativefactors.

Math Tip

In the first part of Step 2, showstudents that the order in whichthey write the addends isimportant. The goal is to have thefirst two terms and the last twoterms have a common factor.

Error Prevention

Some students may look for anynumbers whose sum is -17.Remind them that only the factorpair in the table with a sum of -17 can be used to factor theexpression.

Math Tip

Placement of the signs is veryimportant. To emphasize this,have students switch the signs inthe binomial factors and multiply.

EXAMPLEEXAMPLE44

EXAMPLEEXAMPLE33

EXAMPLEEXAMPLE22

Advanced LearnersChallenge students to use a spreadsheet program tocreate tables like those used in the examples forfinding the sum of possible factors.

English Language Learners ELLMake sure students understand the meaning of theterms factoring and greatest common factor. Useinteger examples and algebraic expressions toillustrate the meaning of each term.

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Factor x2 + 8x + 7.




x2 + 8x + 7



x(x + 1) + 7(x + 1)


Check (x + 1)(x + 7) = x2 + 7x + x + 7

= x2 + 8x + 7 ✓




Factor x2 - 17x + 72.




x2 - 17x + 72




!1, !72!73

!2, !36!38

!3, !24!27

!4, !18 !22

!6, !12!18

!8, !9!17

EXAMPLEEXAMPLE33

22Quick Check


1, 78


EXAMPLEEXAMPLE22



(x ± 2)(x ± 4) (x ± 4)(x ± 8) (x ± 4)(x ± 10)

x2 x x x x x x x

x 1 1 1 1 1 1 1


Vocabulary Tip

260

2. Teach

Guided Instruction


Math Tip


Error Prevention


Math Tip


EXAMPLEEXAMPLE44

EXAMPLEEXAMPLE33

EXAMPLEEXAMPLE22



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Algebra 2 Chapter 5 Lesson 5-4 Practice 5

Name Class Date

Practice 5-4 Factoring Quadratic Expressions

Factor each expression completely.

1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168




Factor x2 + 8x + 7.




x2 + 8x + 7



x(x + 1) + 7(x + 1)


Check (x + 1)(x + 7) = x2 + 7x + x + 7

= x2 + 8x + 7 ✓




Factor x2 - 17x + 72.




x2 - 17x + 72




!1, !72!73

!2, !36!38

!3, !24!27

!4, !18 !22

!6, !12!18

!8, !9!17

EXAMPLEEXAMPLE33

22Quick Check


1, 78


EXAMPLEEXAMPLE22



(x ± 2)(x ± 4) (x ± 4)(x ± 8) (x ± 4)(x ± 10)

x2 x x x x x x x

x 1 1 1 1 1 1 1


Vocabulary Tip

260

2. Teach

Guided Instruction


Math Tip


Error Prevention


Math Tip


EXAMPLEEXAMPLE44

EXAMPLEEXAMPLE33

EXAMPLEEXAMPLE22



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Factor each expression.a. x2 - 6x + 8 b. x2 - 7x + 12 c. x2 - 11x + 24

Note in Example 3 that the factors of c,-9 and -8, appear in the binomials of thefactored form, (x - 9)(x - 8). That is also the case for the factors in Example 2,and it is the case whenever a = 1. So when a = 1, you can skip a few steps infactoring. See Example 4.

If ac is negative, then the factors of ac have different signs.

Factoring When ac R 0

Factor x2 - x - 12.


Since ac = -12 and b = -1, find factors with product -12 and sum -1.

Step 2 Since a = 1, you can write binomials using the factors you found.

x2 - x - 12

(x - 4)(x + 3) Use the factors you found.

Factor each expression.a. x2 - 14x - 32 b. x2 + 3x - 10 c. x2 + 4x - 5

If ac is positive, as in Examples 2 and 3, then the factors of ac have the same sign.This is true even when a 2 1.

Factoring When a u 1 and ac S 0

Factor 3x2 - 16x + 5.




3x2 - 16x + 5

3x2 - x - 15x + 5 Rewrite bx.(')'* (')'*x(3x - 1) - 5(3x - 1) Find common factors.

(x - 5)(3x - 1) Rewrite using the Distributive Property.

Factor each expression. Check your answers.a. 2x2 + 11x + 12 b. 4x2 + 7x + 3 c. 2x2 - 7x + 6

55Quick Check


!1, !15 !16

!3, !5 !8

EXAMPLEEXAMPLE55

44Quick Check

Factors of –12Sum of factors

1, !12!11

!1, 12 11

2, !6!4

!2, 6 4

3, !4!1

!3, 4 1

EXAMPLEEXAMPLE44

33Quick Check(x – 2)(x – 4) (x – 3)(x – 4) (x – 3)(x – 8)

(x ± 2)(x – 16) (x ± 5)(x – 2) (x ± 5)(x – 1)

(x ± 4)(2x ± 3) (x ± 1)(4x ± 3) (x – 2)(2x – 3)

261

Alternative Method

Have students draw a square withfour sections, such as the oneshown. Then follow these steps:1. Write the first and the last term

as shown.

2. Multiply a and c. Then findfactors of the product thathave a sum of b.4(-15) = -606(-10) = -60

and 6 + (-10) = -43. Write the factors with their

signs and variables in the othertwo boxes. Order does notmatter. Then find the GCF ofeach column and each row. AGCF is negative if both termsare negative.

(2x - 5)(2x + 3)

Additional Examples

Factor each expression.a. 15x2 + 25x + 100 5(3x2 ± 5x ± 20)b. 8m2 + 4m 4m(2m ± 1)

Factor x2 + 10x + 24. (x ± 4)(x ± 6)

Factor x2 - 14x + 33. (x – 3)(x – 11)

Factor x2 + 3x - 28. (x – 4)(x ± 7)

Factor 6x2 - 31x + 35. (3x – 5)(2x – 7)

Factor 6x2 + 11x - 35. (3x – 5)(2x ± 7)

66

55

44

33

22

11

4x2 6x

-10x -15

4x2

-15

EXAMPLEEXAMPLE66

2x

2x ± 3

–5

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168






Factor x2 - x - 12.




x2 - x - 12





Factor 3x2 - 16x + 5.




3x2 - 16x + 5




55Quick Check


!1, !15 !16

!3, !5 !8

EXAMPLEEXAMPLE55

44Quick Check


1, !12!11

!1, 12 11

2, !6!4

!2, 6 4

3, !4!1

!3, 4 1

EXAMPLEEXAMPLE44


(x ± 2)(x – 16) (x ± 5)(x – 2) (x ± 5)(x – 1)

(x ± 4)(2x ± 3) (x ± 1)(4x ± 3) (x – 2)(2x – 3)

261

Alternative Method


as shown.




(2x - 5)(2x + 3)

Additional Examples


Factor x2 + 10x + 24. (x ± 4)(x ± 6)

Factor x2 - 14x + 33. (x – 3)(x – 11)

Factor x2 + 3x - 28. (x – 4)(x ± 7)

Factor 6x2 - 31x + 35. (3x – 5)(2x – 7)

Factor 6x2 + 11x - 35. (3x – 5)(2x ± 7)

66

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4x2 6x

-10x -15

4x2

-15

EXAMPLEEXAMPLE66

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2x ± 3

–5

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 261






Factor x2 - x - 12.




x2 - x - 12





Factor 3x2 - 16x + 5.




3x2 - 16x + 5




55Quick Check


!1, !15 !16

!3, !5 !8

EXAMPLEEXAMPLE55

44Quick Check


1, !12!11

!1, 12 11

2, !6!4

!2, 6 4

3, !4!1

!3, 4 1

EXAMPLEEXAMPLE44


(x ± 2)(x – 16) (x ± 5)(x – 2) (x ± 5)(x – 1)

(x ± 4)(2x ± 3) (x ± 1)(4x ± 3) (x – 2)(2x – 3)

261

Alternative Method


as shown.




(2x - 5)(2x + 3)

Additional Examples


Factor x2 + 10x + 24. (x ± 4)(x ± 6)

Factor x2 - 14x + 33. (x – 3)(x – 11)

Factor x2 + 3x - 28. (x – 4)(x ± 7)

Factor 6x2 - 31x + 35. (3x – 5)(2x – 7)

Factor 6x2 + 11x - 35. (3x – 5)(2x ± 7)

66

55

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33

22

11

4x2 6x

-10x -15

4x2

-15

EXAMPLEEXAMPLE66

2x

2x ± 3

–5

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 261

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168






Factor x2 - x - 12.




x2 - x - 12





Factor 3x2 - 16x + 5.




3x2 - 16x + 5




55Quick Check


!1, !15 !16

!3, !5 !8

EXAMPLEEXAMPLE55

44Quick Check


1, !12!11

!1, 12 11

2, !6!4

!2, 6 4

3, !4!1

!3, 4 1

EXAMPLEEXAMPLE44


(x ± 2)(x – 16) (x ± 5)(x – 2) (x ± 5)(x – 1)

(x ± 4)(2x ± 3) (x ± 1)(4x ± 3) (x – 2)(2x – 3)

261

Alternative Method


as shown.




(2x - 5)(2x + 3)

Additional Examples


Factor x2 + 10x + 24. (x ± 4)(x ± 6)

Factor x2 - 14x + 33. (x – 3)(x – 11)

Factor x2 + 3x - 28. (x – 4)(x ± 7)

Factor 6x2 - 31x + 35. (3x – 5)(2x – 7)

Factor 6x2 + 11x - 35. (3x – 5)(2x ± 7)

66

55

44

33

22

11

4x2 6x

-10x -15

4x2

-15

EXAMPLEEXAMPLE66

2x

2x ± 3

–5

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 261






Factor x2 - x - 12.




x2 - x - 12





Factor 3x2 - 16x + 5.




3x2 - 16x + 5




55Quick Check


!1, !15 !16

!3, !5 !8

EXAMPLEEXAMPLE55

44Quick Check


1, !12!11

!1, 12 11

2, !6!4

!2, 6 4

3, !4!1

!3, 4 1

EXAMPLEEXAMPLE44


(x ± 2)(x – 16) (x ± 5)(x – 2) (x ± 5)(x – 1)

(x ± 4)(2x ± 3) (x ± 1)(4x ± 3) (x – 2)(2x – 3)

261

Alternative Method


as shown.




(2x - 5)(2x + 3)

Additional Examples


Factor x2 + 10x + 24. (x ± 4)(x ± 6)

Factor x2 - 14x + 33. (x – 3)(x – 11)

Factor x2 + 3x - 28. (x – 4)(x ± 7)

Factor 6x2 - 31x + 35. (3x – 5)(2x – 7)

Factor 6x2 + 11x - 35. (3x – 5)(2x ± 7)

66

55

44

33

22

11

4x2 6x

-10x -15

4x2

-15

EXAMPLEEXAMPLE66

2x

2x ± 3

–5

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 261

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168


Again, if ac is negative, then the factors of ac have different signs.

Factoring When a u 1 and ac R 0

Factor 4x2 - 4x - 15.




4x2 - 4x - 154x2 + 6x - 10x - 15 Rewrite bx.(')'* (')'*2x(2x + 3) - 5(2x + 3) Find common factors.

(2x - 5)(2x + 3) Rewrite using the Distributive Property.

Factor each expression.a. 2x2 + 7x - 9 b. 3x2 - 16x - 12 c. 4x2 + 5x - 6

A is the product you obtain when you square a binomial. An example is x2 + 10x + 25, which can be written as (x + 5)2. The first term and the third term of the trinomial are always positive, as they represent thesquares of the two terms of the binomial. The middle term of the trinomial is twotimes the product of the terms of the binomial.

Factoring a Perfect Square Trinomial

Factor 9x2 - 42x + 49.

9x2 - 42x + 49 = (3x)2 - 42x + 72 Rewrite the first and third terms as squares.

= (3x)2 - 2(3x)(7) + 72 Rewrite the middle term to verify the perfect square trinomial pattern.

= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2

Factor each expression.a. 4x2 + 12x + 9 b. 64x2 - 16x + 1 c. 25x2 + 90x + 81

77Quick Check

EXAMPLEEXAMPLE77

perfect square trinomial

66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66

12 Factoring Special Expressions

Key Concepts Property Factoring Perfect Square Trinomials

a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip

Point out that you can only usethis formula for the difference oftwo squares. If it was the sum oftwo squares, the last term wouldbe positive. Therefore the signs inthe factors would have to be thesame resulting in a middle termwhen multiplied.

Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2

A square photo is enclosed ina square frame, as shown in thediagram. Express the area of theframe (the shaded area) incompletely factored form. (x ± 7)(x – 7) in.2

Resources• Daily Notetaking Guide 5-4• Daily Notetaking Guide 5-4—

Adapted Instruction

Closure

Ask: What is usually the first step in factoring a quadratictrinomial that is not a perfectsquare trinomial and whose terms have no common factorgreater than 1? Find twointegers that have product acand sum b.

L1

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EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


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x in.7 in.

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EXAMPLEEXAMPLE88

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


L1

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x in.7 in.

88

77

EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


L1

L3

x in.7 in.

88

77

EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


L1

L3

x in.7 in.

88

77

EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


L1

L3

x in.7 in.

88

77

EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262




Factor 4x2 - 4x - 15.









Factor 9x2 - 42x + 49.



= (3x - 7)2 a2 – 2ab ± b2 ≠ (a – b)2


77Quick Check

EXAMPLEEXAMPLE77


66Quick Check


1, !60!59

!1, 60 59

2, !30!28

!2, 30 28

3, !20!17

!3, 2017


4, !15!11

!4, 15 11

5, !12!7

!5, 12 7

6, !10!4

!6, 10 4

EXAMPLEEXAMPLE66



a2 + 2ab + b2 = (a + b)2 a2 - 2ab + b2 = (a - b)2

(x – 1)(2x ± 9) (x – 6)(3x ± 2) (x ± 2)(4x – 3)

(2x ± 3)2 (8x – 1)2 (5x ± 9)2

262

Guided Instruction

Teaching Tip


Additional Examples

Factor 100x2 + 180x + 81. (10x ± 9)2



Adapted Instruction

Closure


L1

L3

x in.7 in.

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77

EXAMPLEEXAMPLE88

PowerPoint

A2_3eTE05_04_259-265 10/20/05 9:27 AM Page 262

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168


An expression of the form a2 - b2 is defined as theIt also follows a pattern that makes it easy to factor.

Multiple Choice The photo shows the thin ring that is the cross-section of the pipe.Which expression gives the area of the cross-section in completely factored form?

p(3 + r)(3 - r) p(3 - r)(3 - r)p(9 - r2) 9 - pr2

Relate equals

minus

Define Let r = inner radius in feet.

Write = -

area = p(3)2 - pr2

= p(32 - r2)

= p(3 + r)(3 - r)

The cross-sectional area is p(3 + r)(3 - r) ft2. The correct choice is A.

Factor each expression: a. x2 - 64 b. 4a2 - 49

Find the GCF of each expression. Then factor the expression.

1. 3a2 + 9 2. 25b2 - 35 3. x2 - 2x

4. 5t2 + 7t 5. 14y2 + 7y 6. 27p2 - 9p


7. x2 + 3x + 2 8. x2 + 5x + 6 9. x2 + 7x + 10

10. x2 + 10x + 16 11. y2 + 15y + 36 12. x2 + 22x + 40

13. x2 - 3x + 2 14. x2 - 13x + 12 15. r2 - 11r + 18

16. x2 - 10x + 24 17. d2 - 12d + 27 18. x2 - 13x + 36

19. x2 - 5x - 14 20. x2 + x - 20 21. x2 - 3x - 40

22. c2 + 2c - 63 23. x2 + 10x - 75 24. t2 - 7t - 44

25. 3x2 + 31x + 36 26. 2x2 - 19x + 24 27. 5r2 + 23r + 26

28. 2m2 - 11m + 15 29. 5t2 + 28t + 32 30. 2x2 - 27x + 36

Example 5(page 261)

Example 4(page 261)

Example 3(page 260)

Example 2(page 260)

Example 1(page 259)

88Quick Check

pr2p(3)2area

the inner area

the outer areapipe’s area

EXAMPLEEXAMPLE Real-World Connection88

difference of two squares.

Practice and Problem SolvingFor more exercises, see Extra Skill and Word Problem Practice.EXERCISES

Practice by ExampleAA

Key Concepts Property Factoring a Difference of Two Squares

a2 - b2 = (a + b)(a - b)

3 ft

r

(x – 8)(x ± 8)

7–30. See margin.

(2a ± 7)(2a – 7)

3; 3(a2 ± 3) 5; 5(5b2 – 7) x; x(x – 2)

t; t(5t ± 7) 7y; 7y(2y ± 1)9p; 9p(3p – 1)GO for

Help

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

A correct solution tothe problem may notbe the correct answerto the question asked.

263

7. (x ± 1)(x ± 2)

8. (x ± 2)(x ± 3)

9. (x ± 2)(x ± 5)

10. (x ± 2)(x ± 8)

11. (y ± 3)(y ± 12)

12. (x ± 2)(x ± 20)

13. (x – 1)(x – 2)

14. (x – 12)(x – 1)

15. (r – 2)(r – 9)

16. (x – 4)(x – 6)

17. (d – 3)(d – 9)

18. (x – 4)(x – 9)

19. (x – 7)(x ± 2)

20. (x ± 5)(x – 4)

21. (x – 8)(x ± 5)

22. (c ± 9)(c – 7)

23. (x ± 15)(x – 5)

24. (t – 11)(t ± 4)

25. (3x ± 4)(x ± 9)

26. (x – 8)(2x – 3)

27. (r ± 2)(5r ± 13)

28. (m – 3)(2m – 5)

29. (t ± 4)(5t ± 8)

30. (x – 12)(2x – 3)

3. PracticeAssignment Guide

A B 1-36, 48-50, 67-70

A B 37-47, 51-66C Challenge 71-78

Test Prep 79-84Mixed Review 85-91

Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 25, 45, 49, 66, 68, 70.

2

1

Guided Problem SolvingGPS

Enrichment

Reteaching

© P

ears

on E

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tion,

Inc.

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right

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serv

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

Practice

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An expression of the form a2 - b2 is defined as theIt also follows a pattern that makes it easy to factor.

Multiple Choice The photo shows the thin ring that is the cross-section of the pipe.Which expression gives the area of the cross-section in completely factored form?

p(3 + r)(3 - r) p(3 - r)(3 - r)p(9 - r2) 9 - pr2

Relate equals

minus

Define Let r = inner radius in feet.

Write = -

area = p(3)2 - pr2

= p(32 - r2)

= p(3 + r)(3 - r)

The cross-sectional area is p(3 + r)(3 - r) ft2. The correct choice is A.

Factor each expression: a. x2 - 64 b. 4a2 - 49

Find the GCF of each expression. Then factor the expression.

1. 3a2 + 9 2. 25b2 - 35 3. x2 - 2x

4. 5t2 + 7t 5. 14y2 + 7y 6. 27p2 - 9p


7. x2 + 3x + 2 8. x2 + 5x + 6 9. x2 + 7x + 10

10. x2 + 10x + 16 11. y2 + 15y + 36 12. x2 + 22x + 40

13. x2 - 3x + 2 14. x2 - 13x + 12 15. r2 - 11r + 18

16. x2 - 10x + 24 17. d2 - 12d + 27 18. x2 - 13x + 36

19. x2 - 5x - 14 20. x2 + x - 20 21. x2 - 3x - 40

22. c2 + 2c - 63 23. x2 + 10x - 75 24. t2 - 7t - 44

25. 3x2 + 31x + 36 26. 2x2 - 19x + 24 27. 5r2 + 23r + 26

28. 2m2 - 11m + 15 29. 5t2 + 28t + 32 30. 2x2 - 27x + 36

Example 5(page 261)

Example 4(page 261)

Example 3(page 260)

Example 2(page 260)

Example 1(page 259)

88Quick Check

pr2p(3)2area

the inner area

the outer areapipe’s area


difference of two squares.

Practice and Problem SolvingFor more exercises, see Extra Skill and Word Problem Practice.EXERCISES

Practice by ExampleAA

Key Concepts Property Factoring a Difference of Two Squares

a2 - b2 = (a + b)(a - b)

3 ft

r

(x – 8)(x ± 8)

7–30. See margin.

(2a ± 7)(2a – 7)

3; 3(a2 ± 3) 5; 5(5b2 – 7) x; x(x – 2)

t; t(5t ± 7) 7y; 7y(2y ± 1)9p; 9p(3p – 1)GO for

Help

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

A correct solution tothe problem may notbe the correct answerto the question asked.

263

7. (x ± 1)(x ± 2)

8. (x ± 2)(x ± 3)

9. (x ± 2)(x ± 5)

10. (x ± 2)(x ± 8)

11. (y ± 3)(y ± 12)

12. (x ± 2)(x ± 20)

13. (x – 1)(x – 2)

14. (x – 12)(x – 1)

15. (r – 2)(r – 9)

16. (x – 4)(x – 6)

17. (d – 3)(d – 9)

18. (x – 4)(x – 9)

19. (x – 7)(x ± 2)

20. (x ± 5)(x – 4)

21. (x – 8)(x ± 5)

22. (c ± 9)(c – 7)

23. (x ± 15)(x – 5)

24. (t – 11)(t ± 4)

25. (3x ± 4)(x ± 9)

26. (x – 8)(2x – 3)

27. (r ± 2)(5r ± 13)

28. (m – 3)(2m – 5)

29. (t ± 4)(5t ± 8)

30. (x – 12)(2x – 3)

3. PracticeAssignment Guide

A B 1-36, 48-50, 67-70

A B 37-47, 51-66C Challenge 71-78

Test Prep 79-84Mixed Review 85-91

Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 25, 45, 49, 66, 68, 70.

2

1

Guided Problem SolvingGPS

Enrichment

Reteaching

© P

ears

on E

duca

tion,

Inc.

All

right

s re

serv

ed.

Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

Practice

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1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

© P

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

© P

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Name Class Date



1. x2 + 4x + 4 2. x2 - 7x + 10 3. x2 + 7x - 8

4. x2 - 6x 5. 2x2 - 9x + 4 6. x2 + 2x - 35

7. x2 + 6x + 5 8. x2 - 9 9. x2 - 13x - 48

10. x2 - 4 11. 4x2 + x 12. x2 - 29x + 100

13. x2 - x - 6 14. 9x2 - 1 15. 3x2 - 2x

16. x2 - 64 17. x2 - 25 18. x2 - 81

19. x2 - 36 20. x2 - 100 21. x2 - 1

22. 4x2 - 1 23. 4x2 - 36 24. 9x2 - 4

25. x2 - 7x - 8 26. x2 + 13x + 36 27. x2 - 5x + 6

28. x2 + 5x + 4 29. x2 - 21x - 22 30. x2 + 13x + 40

31. 2x2 - 5x - 3 32. x2 + 10x - 11 33. x2 - 14x + 24

34. 5x2 + 4x - 12 35. 2x2 - 5x - 7 36. 2x2 + 13x + 15

37. 3x2 - 7x - 6 38. 3x2 + 16x + 21 39. x2 + 5x - 24

40. x2 + 34x - 72 41. x2 - 11x 42. 3x2 + 21x

43. x2 + 8x + 12 44. x2 - 10x + 24 45. x2 + 7x - 30

46. x2 - 2x - 168 47. x2 - x - 72 48. 4x2 - 25

49. x2 - 121 50. x2 + 17x + 16 51. 10x2 - 17x + 3

52. 4x2 + 12x + 9 53. 4x2 - 4x - 15 54. 9x2 - 4

55. x2 + 6x - 40 56. 2x2 - 8 57. x2 + 18x + 77

58. 2x2 - 98 59. x2 + 21x + 98 60. x2 + 20x + 84

61. 9x2 + 30x + 16 62. 8x2 - 6x - 27 63. x2 - 3x - 54

64. x2 - 169 65. 25x2 - 9 66. 7x2 + 49

67. 2x2 - 10x - 28 68. x2 + 8x + 12 69. x2 - 2x - 35

70. x2 + 2x - 63 71. 20x2 - 11x - 3 72. 12x2 + 4x - 5

73. 4x2 - 5x - 6 74. 8x2 + 22x - 21 75. 3x2 - 3x - 168

Name_____________________________________ Class____________________________ Date________________

Lesson ObjectivesSolving quadratic equations by factoringand finding square rootsSolving quadratic equations by graphing2

1

NAEP 2005 Strand: AlgebraTopic: Equations and Inequalities

Local Standards: ____________________________________

Vocabulary and Key Concepts.

Zero-Product PropertyIf ab ! 0, then a ! 0 or b ! 0.Example If (x " 3)(x # 7) ! 0, then (x " 3) ! 0 or (x # 7) ! 0.

The standard form of a quadratic equation is

A zero of a function is

Examples.

Solving by Factoring Solve 3x2 # 20x # 7 ! 0.

3x2!

!

# #20x 7

3x2#

3x

x

"

"

#x 7

0

3x2# !20x

#

"

203

7 3x2

3

# !20x

#

#

20

7

! 7

! 7

! 7

! 7

0

! 0

Write in standard form.

Rewrite term.bx

Find the common factors.

Factor using the Property.

( ) ( )! 0( )( )! 0!

! x !

0

The solutions are and .

Check

or

or

( )2 ( )

! 7 ✓

! 7 ✓

( )2 ( )

Use the Property.

Solve for x.

1

Lesson 5-5 Quadratic Equations

99Daily Notetaking Guide Algebra 2 Lesson 5-5

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Lesson 5-5 Quadratic Equations 267

Quadratic Equations

The is ax2 + bx + c = 0, where a 2 0.You can solve some quadratic equations in standard form by factoring thequadratic expression and then using the Zero-Product Property.

Solving by Factoring

Solve 2x2 - 11x = -15.

2x2 - 11x + 15 = 0 Write in standard form.

(x - 3)(2x - 5) = 0 Factor the quadratic expression.

x - 3 = 0 or 2x - 5 = 0 Use the Zero-Product Property.

x = 3 or x = Solve for x.

The solutions are 3 and .

Check 2x2 - 11x = -15 2x2 - 11x = -15

2(3)2 - 11(3) 0-15 - 0-15

18 - 33 0-15 - 0-15

-15 = -15 ✓ -15 = -15 ✓

Solve each equation by factoring. Check your answers.a. x2 + 7x = 18 b. 2x2 + 4x = 6 c. 16x2 = 8x

11Quick Check

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EXAMPLEEXAMPLE11

standard form of a quadratic equation

5-55-5

Lessons 5-2 and 5-4

Factor each expression.1. x2 + 5x - 14 2. 4x2 - 12x 3. 9x2 - 16

Graph each function.4. y = x2 - 2x - 5 5. y = x2 - 4x + 4 6. y = x2 - 4x

New Vocabulary • standard form of a quadratic equation • Zero-Product Property • zero of a function

What You’ll Learn• To solve quadratic

equations by factoring andby finding square roots

• To solve quadraticequations by graphing

. . . And WhyTo solve equations involvingart, as in Example 5

11 Solving by Factoring and Finding Square Roots

Key Concepts Property Zero-Product Property

If ab = 0, then a = 0 or b = 0.

Example If (x + 3)(x - 7) = 0, then (x + 3) = 0 or (x - 7) = 0.

(x ± 7)(x – 2) 4x(x – 3)4–6. See margin p. 269.

(3x – 4)(3x ± 4)


–9, 2 –3, 1 0, 12

1. PlanObjectives1 To solve quadratic equations

by factoring and by findingsquare roots

2 To solve quadratic equationsby graphing

Examples1 Solving by Factoring2 Solving by Finding Square

Roots3 Real-World Connection4 Solving by Tables5 Solving by Graphing

Math Background

The use of the Zero-ProductProperty allows students to solvequadratic equations that can befactored. Other useful techniquesinclude finding square roots andgraphing. However, each of thesetechniques has limitations. Thisshould begin to convince studentsof the need for a method withoutlimitations. Such a method ispresented in Lesson 5-8.

More Math Background: p. 236D





Properties of ParabolasLesson 5-2: Example 2Extra Skills and Word


Factoring Quadratic ExpressionsLesson 5-4: Examples 2, 3Extra Skills and Word


5-55-5

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Special NeedsHelp students recognize the different methods forsolving quadratic equations. Emphasize that tablesand graphs give approximate solutions, whilealgebraic solutions are exact.

Below LevelRemind students that finding square roots yields twopossible solutions to an equation. x2 = 4 has twosolutions, x = 42.

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learning style: verbal learning style: verbal

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Quadratic Equations



Solve 2x2 - 11x = -15.






Check 2x2 - 11x = -15 2x2 - 11x = -15

2(3)2 - 11(3) 0-15 - 0-15

18 - 33 0-15 - 0-15

-15 = -15 ✓ -15 = -15 ✓


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252

11Q52R2Q52R252

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EXAMPLEEXAMPLE11


5-55-5

Lessons 5-2 and 5-4










If ab = 0, then a = 0 or b = 0.


(x ± 7)(x – 2) 4x(x – 3)4–6. See margin p. 269.

(3x – 4)(3x ± 4)


–9, 2 –3, 1 0, 12






Math Background











5-55-5

267

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Quadratic Equations



Solve 2x2 - 11x = -15.






Check 2x2 - 11x = -15 2x2 - 11x = -15

2(3)2 - 11(3) 0-15 - 0-15

18 - 33 0-15 - 0-15

-15 = -15 ✓ -15 = -15 ✓


11Quick Check

552

252

11Q52R2Q52R252

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EXAMPLEEXAMPLE11


5-55-5

Lessons 5-2 and 5-4










If ab = 0, then a = 0 or b = 0.


(x ± 7)(x – 2) 4x(x – 3)4–6. See margin p. 269.

(3x – 4)(3x ± 4)


–9, 2 –3, 1 0, 12






Math Background











5-55-5

267

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Quadratic Equations



Solve 2x2 - 11x = -15.






Check 2x2 - 11x = -15 2x2 - 11x = -15

2(3)2 - 11(3) 0-15 - 0-15

18 - 33 0-15 - 0-15

-15 = -15 ✓ -15 = -15 ✓


11Quick Check

552

252

11Q52R2Q52R252

52

EXAMPLEEXAMPLE11


5-55-5

Lessons 5-2 and 5-4










If ab = 0, then a = 0 or b = 0.


(x ± 7)(x – 2) 4x(x – 3)4–6. See margin p. 269.

(3x – 4)(3x ± 4)


–9, 2 –3, 1 0, 12






Math Background











5-55-5

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Practice 5-5 Quadratic Equations

Solve each equation by factoring, by taking square roots, or by graphing.When necessary, round your answer to the nearest hundredth.

1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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You can solve an equation in the form ax2 = c by finding square roots.

Solving by Finding Square Roots

Solve 5x2 - 180 = 0.

5x2 - 180 = 0

5x2 = 180 Rewrite in the form ax2 ≠ c.

= Isolate x2.

x2 = 36 Simplify.

x = 46 Find square roots.

Solve each equation by finding square roots.a. 4x2 - 25 = 0 b. 3x2 = 24 c. x2 - = 0

Firefighting Smoke jumpers are in free fall from the time they jump out of a planeuntil they open their parachutes. The function y =-16t2 + 1600 models a jumper’sheight y in feet at t seconds for a jump from 1600 ft. How long is a jumper in freefall if the parachute opens at 1000 ft?

y = -16t2 + 1600

1000 = -16t2 + 1600 Substitute 1000 for y.

-600 = -16t2 Isolate t2.

37.5 = t2

46.1 < t Find square roots.

The jumper is in free fall for about 6.1 seconds.

Check Is the answer reasonable? The negative number -6.1 is also a solution tothe equation. However, since a negative value for time has no meaning inthis case, only the positive solution is reasonable.

a. A smoke jumper jumps from 1400 ft. The function describing the height is y = -16t2 + 1400. Using square roots, find the time during which the jumper is in free fall if the parachute opens at 1000 ft.

b. Solve the equation in part (a) by factoring. Which method do you prefer—usingsquare roots or factoring? Explain.

Not every quadratic equation can be solved by factoring or by finding square roots.You can solve ax2 + bx + c = 0 by graphing y1 = ax2 + bx + c— its relatedquadratic function. The value of y1 is 0 where the graph intersects the x-axis. Eachx-intercept is a and a root of the equation.

You can also solve ax2 + bx + c = 0 by displaying values of y1 = ax2 + bx + cin a table. Scroll through the table to find where y1 changes sign, effectively wherethe graph crosses the x-axis. Then “zoom-in” on the y1 values by adjusting TblStartand ∆ Tbl.

zero of the function

33Quick Check


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5x25

EXAMPLEEXAMPLE22

12 Solving by Graphing

ConnectionReal-World

Careers Smoke jumpers arefirefighters who parachuteinto areas near forest fires.

– , 5252 –2 , 2"2"2 – , 12

12

5 s

t ≠ 5 or t ≠ –5, and use positivesolution because it describes time; check students’ work.

268

2. Teach

Guided Instruction

Connection to Logic

Point out that the word or is theappropriate word to use whenthe Zero-Product Property isbeing used. This is so, because x = 3 and x = cannot simulta-neously be true. On the otherhand, the word and is theappropriate word when you arelisting the solutions, because eachof the numbers will make theoriginal equation true.

Careers

This example demonstrates justhow important Algebra is tosmoke jumpers or any otherskydivers. Without equations suchas these, they would not knowwhen to release their parachutes,possibly endangering their lives.

Additional Examples

Solve 3x2 - 20x - 7 = 0. , 7

Solve 6x2 - 486 = 0. w9

The function y = -16x2 + 270models the height y in feet of aheavy object x seconds after it isdropped from the top of abuilding that is 270 feet tall. Howlong does it take the object to hitthe ground? about 4.1 s

33

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EXAMPLEEXAMPLE11

Advanced LearnersHave students research the trajectory of a specificprojectile, and graph the equation.

English Language Learners ELLAsk students what the term similar means in everydaylanguage. Use this definition to help studentsunderstand the geometric meaning of similar figuresas having the same shape (angles) but different size(sides are proportionally related).

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Solve 5x2 - 180 = 0.

5x2 - 180 = 0


= Isolate x2.

x2 = 36 Simplify.




y = -16t2 + 1600


-600 = -16t2 Isolate t2.

37.5 = t2









33Quick Check


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5x25

EXAMPLEEXAMPLE22




– , 5252 –2 , 2"2"2 – , 12

12

5 s


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2. Teach

Guided Instruction

Connection to Logic


Careers


Additional Examples

Solve 3x2 - 20x - 7 = 0. , 7

Solve 6x2 - 486 = 0. w9


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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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1. x2 - 18x - 40 = 0 2. 16x2 = 56x 3. 5x2 = 15x

4. x2 - 6x - 7 = 0 5. x2 - 49 = 0 6. x2 + 2x + 1 = 0

7. x2 - 1 = 0 8. x2 - 3x - 4 = 0 9. x2 + 9x2 + 20 = 0

10. 6x2 + 9 = -55x 11. (x + 5)2 = 36 12. 2x2 - 3x = 0

13. 2x2 + x - 10 = 0 14. -4x2 + 3x = -1 15. 5x2 - 6x + 1 = 0

16. 3x2 + 1 = -4x 17. -2x2 + 2 = -3x 18. 6x2 + 1 = 5x

19. -2x2 - x + 1 = 0 20. 3x2 + 5x = 2 21. x2 - 6x = -8

22. x2 + 6 = -7x 23. 6x2 + 18x = 0 24. 2x2 + 5 = 11x

25. 3x2 - 7x + 2 = 0 26. 2x2 - 3x = -1 27. 2x2 - x = 6

28. x2 - 144 = 0 29. 4x2 + 2 = 6x 30. 5x2 + 2 = -7x

31. 7x2 + 6x - 1 = 0 32. 2x2 - 6x = -4 33. 11x2 - 12x + 1 = 0

34. 7x2 + 1 = -8x 35. x2 + 9 = -10x 36. (x - 2)2 = 18

37. x2 - 8x + 7 = 0 38. x2 - 16 = 0 39. x2 + 6x = -8

40. x2 + 3 = 4x 41. 2x2 + 6 = -7x 42. 6x2 + 2 = 7x

43. (x + 7)2 = 44. 9x2 - 8x = 1 45. 10x2 + 7x + 1 = 0

46. 4x2 + 2 = -9x 47. 3x2 + 4 = 8x 48. 4x2 + 5 + 9x = 0

49. 9x2 + 10x = -1 50. 2x2 + 9x + 4 = 0 51. 2x2 + 6x = -4

52. 11x2 - 1 = -10x 53. 4x2 = 1 54. 6x2 = 12x

55. 25x2 - 9 = 0 56. 2x2 + 11x = 6 57. 8x2 - 6x + 1 = 0

58. x2 + 11 = -12x 59. 6x2 + 2 = 13x 60. x2 = 121

61. 4x2 - 11x = 3 62. 8x2 + 6x + 1 = 0 63. x2 + 9x + 8 = 0

64. x2 + 8x = -12 65. x2 + 6x = 40 66. 2x2 = 8

67. x2 = x + 6 68. x2 + 2x - 6 = 0 69. x2 - 12 = 0

70. 3x2 + 4x = 6 71. 7x2 - 105 = 0 72. 16x2 = 81

73. x2 + 5x + 4 = 0 74. x2 + 36 = -13x 75. x2 + 6 = 5x

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Solve 5x2 - 180 = 0.

5x2 - 180 = 0


= Isolate x2.

x2 = 36 Simplify.




y = -16t2 + 1600


-600 = -16t2 Isolate t2.

37.5 = t2









33Quick Check


14

22Quick Check

1805

5x25

EXAMPLEEXAMPLE22




– , 5252 –2 , 2"2"2 – , 12

12

5 s


268

2. Teach

Guided Instruction

Connection to Logic


Careers


Additional Examples

Solve 3x2 - 20x - 7 = 0. , 7

Solve 6x2 - 486 = 0. w9


33

22

21311

EXAMPLEEXAMPLE33

52

EXAMPLEEXAMPLE11



L4


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to ﬁnd common and x rewrite using the distributive ... · of signs of ac and b. l1 l2 learning...

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