1.1e: Exercises - Solve by Factoring

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A: Solve by Factoring

Exercise \(\PageIndex{A}\)

Solve by factoring (Zero Factor Property, GCF, Difference of Squares).

1. \((3a−10)(2a−7)=0 \\[4pt]\)

2. \((5b+1)(6b+1)=0 \\[4pt]\)

3. \(6m(12m−5)=0 \\[4pt]\)

4. \(2x(6x−3)=0 \\[4pt]\)

5. \((2x−1)^2=0 \\[4pt]\)

6. \((3y+5)^2=0 \\[4pt]\)

7. \(8( 2 x + 1 ) ( 3 x - 5 ) = 0 \\[4pt]\)

8. \(4 (5 x - 1 ) ( 2 x + 3 ) = 0 \\[4pt]\)

9. \(x^2 + 4x = 0 \\[4pt]\)

10. \(3x^2 + 2x = 0 \\[4pt]\)

11. \(4x^3 = 36 x^2 \\[4pt]\)

12. \( 16 x^2 = 8x \\[4pt] \)

13. \(7a^2+14a=7a \\[4pt]\)

14. \(2x^2 - 25x = 7x^2 \\[4pt] \)

15. \(49m^2=144 \\[4pt]\)

16. \(625=x^2 \\[4pt]\)

17. \(16y^2=81 \\[4pt]\)

16. \(64p^2=225\\[4pt] \)

19. \(121n^4=36n^2 \\[4pt]\)

20. \(100y^4=9y^2 \\[4pt]\)

Answers to Odd Exercises:

1. \(a=\frac{10}{3},\; a=\frac{7}{2}\)

3. \(m=0,\; m=\frac{5}{12}\)

5. \(x=\frac{1}{2}\)

7. \(x=- \frac { 1 } { 2 } ,\; x=\frac { 5 } { 3 }\)

9. \(x = 0 , \; x =-4 \)

11. \( x = 0,\; x=9 \)

13. \(a=−1,\; a=0\)

15. \(m=\frac{12}{7},\; m=−\frac{12}{7}\)

17. \(y=−\frac{9}{4},\; y=\frac{9}{4}\)

19. \(n=0, \; n=−\frac{6}{11},\; n=\frac{6}{11}\)

\( \bigstar \)

Solve each trinomial equation by factoring.

21. \(x ^ { 4 } - 5 x ^ { 2 } + 4 = 0 \\[4pt]\)

22. \(4 x ^ { 4 } - 37 x ^ { 2 } + 9 = 0 \\[4pt]\)

23. \(x ^ { 2 } - 15 x + 50 = 0 \\[4pt]\)

24. \(x ^ { 2 } + 10 x - 24 = 0 \\[4pt]\)

25. \(x^2+36 = 13x \\[4pt] \)

26. \(n^2=5−6n \\[4pt]\)

27. \(3y^2−18y=−27 \\[4pt]\)

28. \(4x^2 + 100 = 40x \\[4pt] \)

29. \(x^3+36x=12x^2 \\[4pt]\)

30. \(m^3−2m^2=−m \\[4pt]\)

31. \(3y^3+48y=24y^2 \\[4pt]\)

32. \(2y^3+2y^2=12y \\[4pt] \)

33. \((x+6)(x−3)=−8 \\[4pt]\)

34. \((p−5)(p+3)=−7 \\[4pt]\)

35. \(( x + 4 ) ( x - 2 ) = 16 \\[4pt]\)
36. \(( x + 1 ) ( x - 7 ) = 9 \\[4pt]\)

37. \((x+1)(x−3)=−4x \\[4pt]\)

38. \((y−3)(y+2)=4y \\[4pt]\)

39. \(( x + 2 ) ( x - 8 ) = 2(x - 14 )\\[4pt]\)

40. \((x+4)(x-6) = 2(x+4) \\[4pt] \)

Answers to Odd Exercises:

21. \( x=\pm 1 ,\; x= \pm 2\)

23. \(x=5, \; x=10\)

25. \(a=4,\; a=9\)

27. \(y = 3\)

29. \( x=0 \; x=6\)

31. \(x=0,\space x=4\)

33. \(x=2,\; x=−5\)

35. \( x=- 6, \; x=4\)

37. \(x=1,\; x=−3\)

39. \( x=2, \; x=6\)

\( \bigstar \)

Solve each trinomial equation by factoring.

41. \(3 x ^ { 2 } + 2 x - 5 = 0 \\[4pt]\)

42. \(2 x ^ { 2 } + 9 x + 7 = 0 \\[4pt]\)

43. \(5a^2−26a=24 \\[4pt]\)

44. \(4b^2+7b=−3 \\[4pt]\)

45. \(4m^2=17m−15 \\[4pt]\)

46. \(4x^2−13x=−3 \\[4pt]\)

47. \(20x^2−60x=−45 \\[4pt]\)

48. \(18b^2+60b+50=0 \\[4pt]\)

49. \(15x^2−10x=40 \\[4pt]\)

50. \(14y^2−77y=−35 \\[4pt]\)

51. \(18x^2−9=−21x \\[4pt]\)

52. \(16y^2+12=−32y \\[4pt]\)

53. \(6 x ^ { 2 } - 5 x - 2 = 30 x + 4 \\[4pt]\)

54. \(6 x ^ { 2 } - 9 x + 15 = 20 x - 13 \\[4pt]\)

55. \(5x^2 - 39x + 12 = 4(x -3) \\[4pt]\)

56. \(4 x ^ { 2 } + 5 x - 5 = 15 ( 3 - 2 x ) \\[4pt]\)

57. \(( 6 x + 1 ) ( x + 1 ) = 6 \\[4pt]\)

58. \(( 2 x - 1 ) ( x - 4 ) = 39 \\[4pt]\)

59. \((3x−2)(x+4)=12x \\[4pt]\)

60. \((2y−3)(3y−1)=8y \\[4pt]\)

Answers to Odd Exercises:

41. \( x=- \frac { 5 } { 3 } , \; x= 1\)

43. \(a=−\frac{4}{5},\; a=6\)

45. \(m=\frac{5}{4},\; m=3\)

47. \(x=\frac{3}{2}\)

49. \(x=2,\; x=−\frac{4}{3}\)

51. \(x=−\frac{3}{2},\; x=\frac{1}{3}\)

53. \( x=- \frac { 1 } { 6 } , \; x=6\)

55. \( x=\frac { 3 } { 5 } , \; x=8\)

57. \( x=- \frac { 5 } { 3 } , \; x=\frac { 1 } { 2 }\)

59. \( x=- \frac { 4 } { 3 } , \; x=2\)

\( \bigstar \)

Solve each polynomial equation by factoring.

61. \(4 x ^ { 3 } - 14 x ^ { 2 } - 30 x = 0 \\[4pt]\)

62. \(9 x ^ { 3 } + 48 x ^ { 2 } - 36 x = 0 \\[4pt]\)

63. \(- 10 x ^ { 3 } - 28 x ^ { 2 } + 48 x = 0 \\[4pt]\)

64. \(- 2 x ^ { 3 } + 15 x ^ { 2 } + 50 x = 0 \\[4pt]\)

65. \(16p^3=24p^2-9p \\[4pt]\)

66. \(36x^3+24x^2=−4x \\[4pt] \)

67. \(\dfrac { 1 } { 10 } x ^ { 2 } - \dfrac { 7 } { 15 } x - \dfrac { 1 } { 6 } = 0 \\[4pt]\)

68. \(\dfrac { 2 } { 3} x ^2 - \dfrac { 7 } { 3} x - \dfrac { 5 } {4 } = 0 \\[4pt]\)

69. \(\dfrac { 1 } { 4 } - \dfrac { 4 } { 9 } x ^ { 2 } = 0 \\[4pt]\)

70. \( \dfrac { 2 } {25 } x ^ { 2 } = \dfrac { 1 } {2 } \\[4pt]\)

71. \(\dfrac { 1 } { 3 } x ^ { 3 } - \dfrac { 3 } { 4 } x = 0 \\[4pt]\)

72. \(\dfrac { 1 } { 2 } x ^ { 3 } - \dfrac { 1 } { 50 } x = 0 \\[4pt] \)

73. \(2 x ^ { 3 } - x ^ { 2 } - 72 x + 36 = 0 \\[4pt]\)

74. \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0 \\[4pt]\)

75. \(45 x ^ { 3 } - 9 x ^ { 2 } - 5 x + 1 = 0 \\[4pt]\)

76. \(4 x ^ { 3 } - 32 x ^ { 2 } - 9 x + 72 = 0 \\[4pt]\)

77. \(16x^2 +24x +9 = 144x^2 \\[4pt]\)

78. \(25x^2 - 80x + 64 = 100x^2 \\[4pt]\)

79. \(9x^2 +12x +4 = 144 \\[4pt]\)

80. \(16x^2-40x+25 = 36 \\[4pt]\)

Answers to Odd Exercises:

61. \( x=0 , \; x=- \frac { 3 } { 2 } , \; x=5\)

63. \( x=- 4, \;x=0 , \; x=\frac { 6 } { 5 }\)

65. \(p=0,\; p=\frac{3}{4}\)

67. \( x=-\frac { 1 } { 3 } , \; x=5\)

69. \( x = \pm \frac{3}{4} \)

71. \( x=0 ,\; x= \pm \frac { 3 } { 2 }\)

73. \( x=\pm 6 ,\; x= \frac { 1 } { 2 }\)

75. \( x=\pm \frac { 1 } { 3 } , \;x= \frac { 1 } { 5 }\)

77. \(x=3/8, \; x=-3/16 \)

79. \(x=10/3, \; x = -14/3 \)

\( \bigstar \)