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(1)

12.1 Solving Quadratic Equations by Factoring

1. x = 2, –4 2. y = –5, –2 3. z = 0, –7 4. x = 0, –12 5. x = 0, –1, 10 6. a = 0, 1, –2

3

7. b = –1

4, –

2 3

8. x = 11

2 ,

5 4

9. y = –3

(2)

21. {–4, 6} 22. {–6, 2} 23. {±13}

24. 3

4

⎧ ⎫ ± ⎨ ⎬ ⎩ ⎭

25. 3

2

⎧ ⎫ ± ⎨ ⎬ ⎩ ⎭

26. {–10, –5} 27. {–3, 24} 28. 3, 1

4 2

⎧ ⎫

− −

⎨ ⎬

⎩ ⎭

29. 3 4, 2 5

⎧ ⎫

−

⎨ ⎬

⎩ ⎭

30. 4 3, 7 2

⎧ ⎫

⎨ ⎬

⎩ ⎭

31. {–2, 0, 1} 32. {0, ±2}

33. 15 and 16 or –16 and –15 34. 2 and 3 or 7 and 8

35. 3 ft

36. 2 and 8 37. 14

38. 8 ft; 20 ft × 28 ft 39. 6, 0, 7

5 3

⎧ ⎫

−

⎨ ⎬

(3)

40. {–8, –3, 2} 41. ≈ 9.8 ft/sec 42. d =

2

v g

43. x = 2, 6 44. ∅

45. x = –9, –5 46. k = 10, 22

47. k = 7, 8, 13 48. k = 1, 4, 11 49. 6 3

50. 57

51. 2 33

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12.2 Solving Quadratic Equations by Taking Roots

1. x = ±5 2. x = ±11 3. x = ± 6

4. x = ± 15

5. x = ±10 6. x = ±4 7. x = ±6 8. x = ±12

9. no real solution 10. x = ± 5

11. x = ± 17

12. no real solution 13. x = ±3 3

14. x = ±2 6

15. x = ±5 2 2

16. x = ±2 15 3 17. x = ±5 18. x = ±2 5

(5)

20. x = 6 ± 13

21. x = –2 ± 2 3

22. x = 1, –9 23. x = 8, –14

24. no real solution 25. x = –2

26. x = ±3

2

27. x = 1

2, –1

28. x = 2

3, –

2 9

29. x = 1

2, –

5 2

30. x = 10

3 ,

2 3

31. 12 ft 32. 22 ft 33. 64 ft 34. 2.7 sec

35. ≈ 3.5 sec; 5 sec;

The object continues to fall faster as time progresses. 36. x = 6 14

10

(6)

Section 12.2

37. x = 5 4 2

2

±

38. 25 ft

39a. d = |hc – hi| or d = |hi – hc| 39b. hc = –

1

2gt

2

+ hi 39c. hc = –16t2 + hi

39d. 9.8 m/sec2; hc = –4.9t2 + hi 40a. 65 ft

40b. 2.0 sec

41a. a ≈ –4.06;

h(x) = –4.06x2 + 65 41b. a ≈ –2.15;

h(x) = –2.15x2 + 65 41c. a ≈ –2.88;

h(x) = –2.88x2 + 65 41d. Answers will vary;

[0, 10] × [0, 70] is reasonable.

(7)

Section 12.2

45.

2

1 3

2 8

⎛ _× ⎞

⎜ ⎟

⎝ ⎠ ;

9 256

46. x = –3, 17 47. x = ±8

3

48. x = ±2, ±3 49. x = ±2, –1

2

50. $892.73 at 5% and $922.73 at 6%

(8)

12.3 Completing the Square

1. 16

2. 36

3. 64

4. 25

5. 1

6. 676

7. 9

4

8. 25

4

9. 121

4

10. 81

4

11. 9

16

12. 25

144

13. 25

121

14. 4

25

(9)

18. y = 7, –1 19. {–2, 12} 20. {–11, 5} 21. {–9, 11} 22. {4, 10} 23. m = 4

3, 0 24. x = 0, –5

6 25. x = 1, –2 26. a = 5, 2

27. x = –10 ± 85 ≈ –0.78, –19.22

28. x = 6 ± 41 ≈ 12.40, –0.40 29. x = 9 ± 6 2 ≈ 17.49, 0.51 30. x = 4 ± 3 2 ≈ 8.24, –0.24 31. x = 4 ± 2 3 ≈ 7.46, 0.54 32. x = 6 ± 2 10 ≈ 12.32, –0.32 33. x = 2 ± 7 ≈ 4.65, –0.65

34. x = 5 ± 3 2 ≈ 9.24, 0.76 35. 10 in. on each side

36. 12 and 18 or –18 and –12 37. x = 3 5

2

(10)

38. m = 5 73 2

− ±

39. 44 ft × 54 ft 40a. 1520 ft2

40b. 36% 41a. 1520 ft2 41b. 28% 42. 37

16

43. –26 49

44. x2 + 3x – 5 45. x2 + 5

3x – 5 46. x2 – 6x + 2 47. x2 – 6x + 2 48. x = 81

49. no real solution 50. x = 25

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_{2012 BJU Press. Unauthorized reproduction prohibited.}

12.4 Completing the Square with Leading Coefficients

1. x = –3 ± 11

2. y = 2 ± 3

3. a = 8, 0

4. b = –1, –11

5. y = –4 ± 19

6. x = –2 ± 2

7. no real solution

8. r = –12 ± 2 34

9. x = 4 ± 2 5

10. no real solution

11. y = 1, –3

2

12. x = 5

2, –2

13. c = 5

2, – 1 2

14. d = 1

3, – 13

3

15. x = 1, –1

2

16. y = 1, –2

(12)

Section 12.4

17. g = 1 17

4 ±

18. h = 3 7

2 ±

20. x = 5 31

2 ±

21. k = 2, –4

3

22. y = –1, –5

3

23. x = –1

2, –1

24. n = 3, –3

2

25. m = 1 2 3

2 ±

26. x = 9 87

3 − ±

27. y = 12 142

2 − ±

28. t = 12 141

3 − ±

29. 11 and 21

30. 11 and 27 or –15 and –25

(13)

Section 12.4

32. 1.5 sec

33. 13 and 25

34. opponent’s 10 yd line

35. opponent’s 13.5 yd line

36. no

37. x = 12 139

5 − ±

38. x = –3 ±₄₃

39. y ≈_{0.56, –3.56; x}≈ ±_0.75

40. 5

41. 42

42. 170 1

64

43. –38

27 or –1

11 27

44. x6

45. 4x2 + 22x + 30

46. 3 14 3

47. 4 6 4

48. 3+ 3

49. 5 5

(14)

12.5 The Quadratic Formula

1. a = 1, b = 6, c = 5 2. a = 2, b = –6, c = 1 3. a = 1, b = 5, c = 6 4. a = 3, b = –2, c = –5 5. a = 1, b = 0, c = –6 6. a = 1, b = –7, c = 0 7. {–4, –2}

8. {–3, –1} 9. {–5, –3} 10. {4, 5} 11. {–3, 3}

12. {1, 8} 13. 1, 3

2

⎧₋ ⎫

⎨ ⎬

⎩ ⎭

14. {–7, 4} 15. {3} 16. 3

2

⎧ ⎫ ⎨ ⎬ ⎩ ⎭

17. x = 4, 1

2

18. x = 5

(15)

19. y = 5 17 4

±

20. w = 7 13 6

±

21. x = 21 3

±

22. x = 5 4, 0 23. x = 5 105

4

±

24. y = 3 137 8

±

25. m = –2

7, –1

26. x = –2 3, –

4 5

27. k = 1 31 3

±

28. no real solution 29. x = 7

2, – 1 4

30. r = 7

5, –1

31. x ≈ 2.39, 0.28 32. y ≈ 1.46, –0.46

(16)

34. Commutative Property of Addition

35. 12 and 18 36. 4 and 30 37. ≈ 4.6 in.

38. 18 in. × 21 in. 39. x ≈ 2.30, 0.66 40. x ≈ 1.65, –0.45 41. y =

2 4 2

s s rt

r

− ± −

42. 108.4 ft 43. yes

44. yes, at about 44 and 76 ft 45. 0 = 2.2x – 22

46. x = 10; y = 3

47. At 10 ft from the first fountainhead the two fountains intersect at a height of 3 ft

(17)

(18)

12.6 More Quadratic Equations

1. a

2. d

3. b

4. a

5. d

6. b

7. 196; a 8. –44; d 9. 0; c 10. 4; a 11. 97; b

12. –32; d 13. 17; b 14. 0; c

15. x = 3, –15; rational 16. q = 9 97

2

− ±

; irrational

17. a = 3 5 2

− ±

; irrational 18. no real solution

19. x = 7 33 4

±

(19)

20. x = 9

4, –1; rational

21. x2 + 4x – 32 = 0 22. x2 – 5x = 0

23. 8x2 – 10x + 3 = 0 24. 6x2 – x – 2 = 0 25. y = 2, 1

6

26. a = 1, –5

3

27. a = –4

3

28. r = 5

4, –1

29. x = –4 ± 3 2 30. t = ±2

31. a = –22 ± 482

32. x = 2 ± 2 33. x = 8

3, –

3 2

34. x = 4, –11

2

(20)

Section 12.6

37. 12 in. × 15 in.

38. 17.9 in, 14.9 in., and 9.9 in. 39. y = –7

6

40. x = 6, 3

41. x ≈ 7.110, –2.110 42. x ≈ 6.364, –0.864 43. 9x2 + 18x + 4 = 0 44. 36x2 + 24x + 1 = 0 45. x = 14

5

46. x = –10

47.

48.

y

x

y

(21)

Section 12.6

49–52.

53. ≈ 8.9 ft 54. 8 in.

y

x

y 5 3uxu _y₅_u_x_u

y 5 23uxu

(22)

12.7 Quadratic Functions: f₍x_{) =}ax2₊c

1. 0, 4, 1, 16

2. 0, –4, –1, –16

3. – 4, 0, –3, 12

4. 5, 9, 6, 21

5. 0, –8, –2, –32

6. 0, 12, 3, 48

7. –7, 5, – 4, 41

(23)

Section 12.7

9–12.

Vertex

Function Direction of Opening

At Minimum or

Maximum

Comparison of Shape of

f(x) = x2

f(x) = 4x2 up (0, 0) minimum steeper

f(x) = –1

3x

2

down (0, 0) maximum flatter

f(x) = –x2 + 4 down (0, 4) maximum same

f(x) = 1

5x

2

(24)

Section 12.7

13.

14.

15.

16. y

x

y

x

y

x

y

(25)

Section 12.7

17.

18.

19.

20. y

x

y

x

y

x

y

(26)

Section 12.7

21.

22.

23.

24. y

x

y

x

y

x

y

(27)

Section 12.7

25.

26.

27.

28. y

x

y

x

y

x

y

(28)

Section 12.7

29.

30.

31a. 125, 150, 175, 200

31b.

31c. 200 ft.

32. origin or (0, 0)

33. (0, 0); (0, 0)

34. y–axis; x =0

y x y x 25 210

25 5

5 10 15 0 50 Appr o xi m ate H ei g h t (ft)

(29)

Section 12.7

35. (0, c)

36. I and II

37. all four

38a. (0, 0.3), (±_{1, 0.5),}

(±_{2, 1.1), and (}±_{3, 2.1)}

38b.

38c. (0, 0.3); minimum

38d. x = 0

39. f(x) = –1.44x2 + 7.2x

40. 5.76 ft, 9 ft, 8.64 ft, and 7.56 ft

41. Multiplication Property of Equality

42. Addition Property of Equality

43. 13

26

44. 7 2 + 15 3

45. 8

46. x = 4

47. x < –3

y

(30)

Section 12.7

48. 109, 111, and 113

49. 131 pennies and 21 dimes

(31)

12.8 Quadratic Function: f(x) = a(x – h)2 + k

1–10.

Function Vertex (h, k)

Line of Symmetry

Value of a

Direction of Opening

Comparison of Shape to f(x) = x2

f(x) = (x –2)2 + 9 (2, 9) x = 2 1 up same

f(x) = – (x + 5)2 + 1 (–5, 1) x = –5 –1 down same

f(x) = 3(x – 2)2 – 4 (2, –4) x = 2 3 up steeper

f(x) = 1

4x

2

– 6 (0, –6) x = 0 1

4

up flatter

f(x) = –5(x + 1)2 (–1, 0) x = –1 –5 down steeper

f(x) = 2

3(x – 8)

2 (8, 0) x = 8 2

3

(32)

f(x) = –7

4x + 7

(0, 7) x = 0 _–7 4

down steeper

f(x) = –6x2 (0, 0) x = 0 –6 down steeper

f(x) = x2 – 8x + 9 (4, –7) x = 4 1 up same

f(x) = x2 – 10x – 7 (5, –32) x = 5 1 up same

11. y 12.

x

y

(33)

13.

14.

15.

16.

y

x

y

x

y

x

y

(34)

17.

18.

19.

20.

y

x

y

x

y

x

y

(35)

21. f(x) = (x + 2)2 – 16; (–2, –16)

22. f(x) = (x – 3)2 – 1; (3, –1) 23. f(x) = (x + 7)2 – 9; (–7, –9) 24. f(x) = (x – 4)2 – 1; (4, –1) 25. (–5, –1);

y

x

26. (2, 1); y

x

27. (– 4, 4);

y

(36)

28. (2, 1); y

x

29. (3, –7); y

x

30. (1, –6); y

x

25

25 5

5 10

31a. $1750 lost; $125 gained 31b. 200 items

31c. $750 32. $400

(37)

34a. 22 – x

34b. f(x) = x(22 – x) or f(x) = –x2 + 22x 34c. (11, 121)

34d. 11 ft × 11 ft 34e. 121 ft2

35a. 90 – 2x

35b. f(x) = x(90 – 2x) or f(x) = –2x2 + 90x 35c. (22.5, 1012.5) 35d. 22.5 ft × 45 ft 35e. 1012.5 ft2

36. (2, –3); y

x

37. (–1, –1.5);

y

(38)

38. 1327 ft; 8 sec 39. (0, 6); (20, 12) 40. a = –0.015

41. f(x) = –0.015(x – 20)2 +12; f(x) = –0.015x2 + 0.6x + 6 42. 48 yd from the initial point

or at the opponent’s 22 yd line

(39)

44. x5 – 1024 45. x + 4 – 11

1 x+

46. 1, 2 2

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

47. 71 and 174 48. x = –1

2

49. x = –15, 7 50. x = –9, 16 51. x = 1, 1

2

52. x = 0, 5

3, –

7 2

(40)

12.9 Zeros of a Function

1. (0, 0) 2. (0, 8) 3. (0, – 5) 4. (0, 0) 5. (0, 1) 6. (0, –3) 7. (0, 8) 8. (0, 4) 9. (0, –5) 10. (0, –3) 11. –2, 4

12. – 4, –2 13. 3

14. –3

15. no real zero 16. no real zero 17. 1, –3

18. 1 ± 2 2 19. –3 ± 17

20. 5 89 4

±

(41)

22. III and IV

23. the x-axis or y = 0 24. one

25. (0, 0); 0; (0, 0);

y

x

26. (0, –4); –2, 2; (0, –4);

y

x

27. (0, 0); –3, 0; 3, – 9 2 4

⎛ ⎞

−

⎜ ⎟

⎝ ⎠;

y

(42)

28. (0, 8); 2, 4; (3, –1);

y

x

29. (0, 6); 2, 3; 5, – 1 2 4

⎛ ⎞

⎜ ⎟

⎝ ⎠ ;

y

x

30. (0, 1); –2 ± 3; (–2, –3); y

x

31. (0, 9); 1.5; (1.5, 0);

y

(43)

32. (0, –4); 2 ± 2 2; (2, –8); y

x

33. (0, –3); no real zero; (–1, –2);

y

x

34. (0, 9.25); no real zero; (5, 3);

y

x

35a. 200 refrigerators 35b. $34,000

(44)

36b. $800 36c. never 37. ah2 + k 38a. s =

4

x

; r =

2

x

π

38b. A_s = 2

16

x

; A_c = 2

4

x

π

38c. the area of the circle; 2 4 x π > 2 16 x

since 4π < 16

39a. 7 121,

3 6

⎛₋ ⎞

⎜ ⎟

⎝ ⎠ or

(

−2.3, 20.16

)

39b. x = 7

3

− or –2.3

39c. downward 39d. steeper 39e. –6, 4

3

(45)

Section 12.9

40.

x y

12.5 7.1 vertex

0 0 zeros

25 0

23 2.1 3 other

points 21 3.9

19 5.2

17 6.2

15 7

41a. f(x) = a(x – 12.5)2 + 7.1 41b. a ≈ –0.045

41c. y = –0.045(x – 12.5)2 + 7.1 42.

43. m = 2; (0, –7); 7

2

44. m = –5

(46)

Section 12.9

45. 29

46. 221

47.

48.

49. x = –2

50. x = 7 109

6

− ±

51. x = 4 ± 2 3

52. x = 5 ± 7

y

x

y

(47)

Chapter 12 Review

1. x = 2, –3 2. x = –6, 10 3. a = 0, 1

4, –3

4. x = –5, 2 5. x = –1, 7 6. y = 0, ±1

2

7. x = ± 6

8. y = ±2

9. x = 4 ± 11

10. x = 1 ± 2 3 5 or

5 2 3 5

±

11. x = –1, –7 12. x = 5 37

2

− ±

13. x = 5 ± 110 2 14. t = –3 ± 10

(48)

18. 36; two rational 19. –4; no real

20. 57; two irrational 21. x = 3 ± 7

22. x = 7, –3 23. x = 1, –5

3

24. x = 3

2. –1

25. {–2, 7} 26. {±4} 27. {±3}

29. 1 21

2

⎧ _± ⎫ ⎪

⎨ ⎬

⎪

⎩ ⎭

30. 0, 8 5 ⎧ ⎫ ⎨ ⎬ ⎩ ⎭

31. {–6, –3} 32. {–3}

33. –7 and 12 or –12 and 7 34. 7 and 16 or –14 and –8 35. 15 ft × 8 ft

(49)

38. –2 39. –3 40. 2, 4 41. 1; x = 3 42. x = 3 43. (0, 0); 0

y

x

44. (0, –6); ± 6 y

x

45. (2, 0); 2 y

(50)

46. (–6, 2); no real zero y

x

47. (–3, –8); –3 ± 2 2 y

x

48. (1, 4); –1, 3 49. (0, 6)

50. 1, 3

51. (2, –2); minimum 52. yes; The graph opens

upward from the minimum of (2, –2), so it crosses

(51)

Sequences—Quadratic Arithmetic Sums

1. inductive 2. deductive

(Distributive Property) 3. substitution

4. Distributive Property

5. Commutative Property of Addition

6. Distributive Property 7. Sn = 3n2 + 3n

8. Sn = 2

3 3

2

n + n

9. An = 5n – 2 10. Sn =

2 5

2