Foundational to TEKS (4)(F) Solve quadratic and square root equations.

TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Additional TEKS (1)(E)

TEKS FOCUS

•Difference of two squares – an expression of the form a2 - b2

•Factoring – rewriting an expression as the product of its factors

•Greatest common factor (GCF) of an expression – the common factor of each term of the expression that has the greatest coefficient and the greatest exponent

•Perfect square trinomial – a trinomial that is the square of a binomial

•Number sense – the understanding of what numbers mean and how they are related

VOCABULARY

You can factor many quadratic trinomials (ax2 + bx + c) into products of two binomials.

ESSENTIAL UNDERSTANDING

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

Key Concept Factoring Perfect Square Trinomials

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

You can use the FOIL method to multiply two binomials. You can use FOIL in reverse to help you factor.

(x + 4)(x + 2) = x(x) + x(2) + 4(x) + 4(2) F: First; O: Outer; I: Inner; L: Last = x2 + 6x + 8

To factor x2 + 6x + 8, think of FOIL in reverse. Find two binomials for which the first terms have the product x2, the products of the outer and inner terms have the sum 6x, and the last terms have the product 8.

x2 + 6x + 8 = (x + 4)(x + 2)

Key Concept Factoring Using FOIL

5-5 Factoring Quadratic Expressions

178 Lesson 5-5 Factoring Quadratic Expressions

FO

IL

Problem 1

Factoring ax2 ∙ bx ∙ c when a ∙ t1

What is the expression in factored form?

A x2 ∙ 9x ∙ 20

Step 1 Find factors of 20 with sum 9.

Since both 20 and 9 are positive, both factors are positive.

Step 2 Use the factors you found. Write the expression as the product of two binomials.

x2 + 9x + 20 = (x + 4)(x + 5) Use the factors 4 and 5.

B x2 ∙ 14x ∙ 72

Step 1 Find factors of -72 with sum 14.

Since c 6 0, one factor is positive and the other is negative.

Since b 7 0, the factor with greater absolute value is positive.

Step 2 Use the factors you found, -4 and 18. Write x2 + 14x - 72 = (x - 4)(x + 18).

C ∙x2 ∙ 13x ∙ 12

Step 1 Rewrite the expression to show a trinomial with leading coefficient 1.

-(x2 - 13x + 12) Factor out -1.

Step 2 Find factors of 12 with sum -13.

Since c 7 0, both factors have the same sign.

Since b 6 0, both factors must be negative.

Step 3 Use the factors you found, -1 and -12. Write -x2 + 13x - 12 = -(x2 - 13x + 12) = -(x - 1)(x - 12).

TEKS Process Standard (1)(E)

Factors of 20

Sum of factors

1, 20

21

2, 10

12

4, 5

9

Factors of –72

Sum of factors

� � � � � �1, 72

71

2, 36

34

3, 24

21

4, 18

14

6, 12

6

8, 9

1

�1, �12

�13

�2, �6

�8

�3, �4

�7

Factors of 12

Sum of factors

Will factoring out ∙1 change the answer?No, because the final factored expression will include -1 as a factor.

How can you make a table to find factors?Use the first row to list sets of factors of the constant. Use the second row to find the sum of each set of factors.

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Problem 3

Problem 2

Finding Common Factors

What is the expression in factored form?

A 6n2 ∙ 9n

6n2 + 9n = 3n(2n) + 3n(3) Factor out the GCF, 3n.

= 3n(2n + 3) Use the Distributive Property.

B 4x2 ∙ 20x ∙ 56

4x2 + 20x - 56 = 4(x2) + 4(5x) - 4(14) Factor out the GCF, 4.

= 4(x2 + 5x - 14) Use the Distributive Property.

= 4(x - 2)(x + 7) Factor the trinomial.

Factoring ax2 ∙ bx ∙ c when ∣ a ∣ ≠ 1

What is the expression in factored form?

A 2x2 ∙ 11x ∙ 12

Step 1 Since there is no common factor, find ac.

ac = 2(12) = 24

Step 2 Since both b and ac are positive, find positive factors of 24 that have sum 11.

Step 3 Factor the trinomial as follows.

2x2 + 11x + 12

2x2 + 3x + 8x + 12

x(2x + 3) + 4(2x + 3)

(x + 4)(2x + 3)

Rewrite bx. Since ac = 24, try 3 and 8 as coefficients of x.

Find a common factor for the first two terms and another common factor for the last two terms.

Rewrite using the Distributive Property.

B 4x2 ∙ 4x ∙ 3

Step 1 Since there is no common factor, rewrite bx as the sum of two terms with coefficients that are factors of ac, and have sum b.

ac = 4(-3) = -12

Step 2 Since ac = -12 6 0, find factors of ac with opposite signs. Since b 6 0, the factor with greater absolute value is negative.

1, 24

25

2, 12

14

3, 8

11

4, 6

10

Factors of 24

Sum of factors

1, �12

�11

2, �6

�4

3, �4

�1

Factors of –12

Sum of factors

continued on next page ▶

How should you make your table in this case?Use the first row to list sets of factors of ac. Use the second row as before, to find the sum of each set of factors.

Should you factor out a number, a variable, or both?Both; the two terms have numerical and variable common factors.

Do you need positive or negative factors?You need one negative factor and one positive factor.

180 Lesson 5-5 Factoring Quadratic Expressions

Problem 5

How can a binomial be the product of two binomials?If the outer and inner products are opposites, their sum is zero.

Problem 4

Factoring a Difference of Two Squares

What is 25x2 ∙ 49 in factored form?

25x2 - 49 = (5x)2 - 72 Write as the difference of two squares.

= (5x + 7)(5x - 7) Use the pattern for factoring a difference of two squares.

continuedProblem 3

Factoring a Perfect Square Trinomial

What is 4x2 ∙ 24x ∙ 36 in factored form?

TEKS Process Standard (1)(C)

Write the expression.

Use number sense and mental math to factor the expression in one step.

Is the first term a perfect square? Yes, 4x2 is (2x)2.

Is the last term a perfect square? Yes, 36 is 62.

Is the middle term twice the product of 6 and 2x? Yes, 2 ~ 6 ~ 2x = 24x .

There’s a minus sign in the middle. You are done.

4x2 ∙ 24x ∙ 36

(2x ∙ 6)2

(2x ∙ 6)2 ∙ 4x2 ∙ 24x ∙ 36 ✔You should check.

Step 3 Factor the trinomial as follows.

4x2 - 4x - 3

4x2 + 2x - 6x - 3

2x(2x + 1) - 3(2x + 1)

(2x - 3)(2x + 1)

Rewrite bx using b = 2 - 6.

Find a common factor for the first two terms and another common factor for the last two terms.

Rewrite using the Distributive Property.

Check (2x - 3)(2x + 1) = 4x2 + 2x - 6x - 3 = 4x2 - 4x - 3 ✔

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PRACTICE and APPLICATION EXERCISES

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M E W O RK

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Factor each expression.

1. x2 + 3x + 2 2. x2 + 5x + 6 3. x2 + 22x + 40

4. c2 + 2c - 63 5. x2 + 10x - 75 6. - t2 + 7t + 44

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

7. 3a2 + 9 8. 25b2 - 20b 9. x2 - 2x

Factor each expression.

10. 3x2 + 31x + 36 11. 2x2 - 19x + 24 12. 5r2 + 23r + 26

13. 2m2 - 11m + 15 14. 5t2 + 28t + 32 15. 2x2 - 27x + 36

Select Techniques to Solve Problems (1)(C) Use mental math or number sense to help you factor each expression that can be factored. For an expression that cannot be factored into a product of two binomials, explain why.

16. x2 + 2x + 1 17. t2 - 14t + 49 18. k2 - 18k + 81

19. 4z2 - 20z + 25 20. 4x2 + 16x + 8 21. 81z2 + 36z + 4

22. Apply Mathematics (1)(A) Suppose you cut a small square from a square sheet of cardboard. Find the sides of one rectangle whose area is equal to the area of the remaining part.

23. The area in square centimeters of a square area rug is 25x2 - 10x + 1. What are the dimensions of the rug in terms of x?

Factor each expression completely.

24. 9x2 - 36 25. 18z2 - 8 26. 4n2 - 20n + 24

27. 4x2 - 22x + 10 28. -6z2 - 600 29. - 116 s2 + 1

30. a. Multiply (a + b)(a - b)(a2 + b2).

b. Use your result from part (a) to completely factor 81x4 - 256y4.

31. Evaluate Reasonableness (1)(B) Your friend attempted to factor an expression as shown. Find the error in your friend’s work. Then factor the expression correctly.

32. The area in square feet of a rectangular field is x2 - 120x + 3500. The width, in feet, is x - 50. What is the length, in feet?

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

33. y2 - y 34. ab2 - b 35. 10x2 - 90

36. What is the factored form of 4x2 + 15x - 4?

A. (2x + 2)(2x - 2) C. (4x + 1)(x - 4)

B. (2x - 4)(2x + 1) D. (4x - 1)(x + 4)

x

y

2x2 - 7x + 52x2 - 5x - 2x + 5x(2x - 5) + (2x - 5)(x + 1) (2x - 5)

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182 Lesson 5-5 Factoring Quadratic Expressions

TEXAS Test Practice

52. How can you write (m - 5)(m + 4) + 8 as a product of two binomials?

A. (m - 1)(m + 8) C. (m + 8)(m + 8)

B. (m - 4)(m + 3) D. (m - 5)(8m + 32)

53. The graph of a quadratic function has vertex (7, 6). What is the axis of symmetry?

F. x = 6 G. y = 6 H. x = 7 J. y = 7

54. Suppose you hit a baseball and its flight takes a parabolic path. The height of the ball at certain times appears in the table below.

a. Find a quadratic model for the ball’s height as a function of time.

b. Write the quadratic function in factored form.

Time (s)

Height (ft)

0.5

10

0.75

10.5

1

9

1.25

5.5

37. Apply Mathematics (1)(A) What is the volume of the shaded pipe with outer radius R, inner radius r, and height h as shown? Express your answer in completely factored form.

38. Create Representations to Communicate Mathematical Ideas (1)(E) Write a quadratic trinomial that you can factor, where a ≠ 1, ac 7 0, and b 6 0. Factor the expression.

39. Explain Mathematical Ideas (1)(G) Explain how to factor 3x2 + 6x - 72 completely.

Factor each expression completely.

40. 0.25t 2 - 0.16 41. 8100x 2 - 10,000

42. (x + 3)2 + 3(x + 3) - 54 43. (x - 2)2 - 15(x - 2) + 56

44. 6(x + 5)2 - 5(x + 5) + 1 45. 3(2a - 3)2 + 17(2a - 3) + 10

46. Explain Mathematical Ideas (1)(G) Explain how to factor 4x 4 + 24x 3 + 32x 2.

Factor each expression completely.

47. 116 x4 - y4 48. 16x4 - 625y4 49. 243a5 - 3a

50. When the expression x2 + bx - 24 is factored completely, the difference of the factors is 11. Find both factors if it is known that b is negative.

51. Prove that n3 - n is divisible by 3 for all positive integer values of n. (Hint: Factor the expression completely.)

R

h

r

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