1. \(x=36\)
2. \(y = 2\)
3. \(x = -\frac{5}{8}\)
4. \(x = 2\)
5. \(x = 13\)
6. \(z = -4\)
7. \(x = -2\)
8. \(x = 3\)
9. \(x = -5\)
10. \(x = -7\)
11. \(x = 2\)
12. \(x = -4\)
13. \(x = \frac{1}{2}\)
14. \(x = \frac{4}{5}\)
15. \(x = \frac{a}{a + 5}\)
16. \(x = a\)
17. \(x = \frac{b – 1}{2}\)
18. \(x = 2a\)
19. \(x = a + b\)
20. \(x = 2m\)
21. \(x = 2\)
22. \(x = 3b\)
23. \(s = \frac{A + \pi r^2}{\pi r}\)
24. \(a = \frac{ S (I – r) }{1 – r^n}\)
25. \( h = \frac{HS – f}{S}\)
26. \(x = \frac{ay}{y – a}\)
27. \(-2, \; 6\)
28. \(3\)
29. \(-3,\; -8\)
30. \(1, \; \frac{1}{2}\)
31. \(2, \; -\frac{1}{9}\)
32. \(5, \; -\frac{1}{3}\)
33. \(-2, \; -\frac{3}{2}\)
34. \(3, \; -\frac{5}{6}\)
35. \(\frac{1}{4}, \; \frac{1}{6}\)
36. \(8, -\frac{19}{4}\)
37. \(-1, \, 1, \, -4, \, 4\)
38. \(-\frac{1}{2}, \; \frac{1}{2}\)
39. \(-27, \; 8\)
40. \(\frac{1}{8}, \; -8\)
41. \(1 – \frac{ 1 }{3} \sqrt{5}, \, 1 + \frac{1}{3} \sqrt{5}\)
42. \(\frac{1}{2} \sqrt{3}\)
43. \(\frac{3}{4}- \frac{\sqrt{5}}{4}, \, \frac{3}{4} + \frac{\sqrt{5}}{4}\)
44. \(-3 \sqrt{5}, \, 3 \sqrt{5}\)
45. \(-a , \; a + 2\)
46. \(a, \; 2\)
47. \(x = 3\)
48. \(x = 0\)
49. \(x = 0\)
50. \(x = \frac{3}{8}\)
51. \(2\)
52. \(1 – \sqrt{2}, \, 1 + \sqrt{2}\)
53. \(2 – \frac{1}{3} \sqrt{42}, \, 2 + \frac{1}{3}\sqrt{42}\)
54. \(\frac{5}{2}\)
55. \(1 – \sqrt{2}, \, 1 + \sqrt{2}\)
56. \(- 3 – \sqrt{2}, \, – 3 + \sqrt{2}\)
57. \(\frac{1}{3}, \, -\frac{1}{5}\)
58. \(3 , \, \frac{2}{5}\)
59. \(-1, \, \frac{5}{2}\)
60. \(-1\)
61. \(11\)
62. \(-10\)
63. \(10\)
64. \(y = 7\)
65. \(x = 3\)
66. \(z = 9\)
67. \(x = 2\)
68. \(\frac{1}{4}\)
69. \(2\)
70. \(2\)
71. \(4\)
72. \(4\)
73. \(1\)
74. \(2\)
75. \(\frac{9}{16}\)
76. \(2\)
77. \(-4, \; -\frac{2}{5}\)
78. \(4, \, -\frac{11}{13}\)
79. \(-4\)
80. \(-60\)
81. Roots: \(1, \, -2, \, -1; \quad (x – 1)(x + 2)(x + 1)\)
82. Roots: \(1, \, 1 + \sqrt{3}, \, 1 – \sqrt{3}\); \((x – 1) \left( x – 1 – \sqrt{3} \right) \left( x – 1 + \sqrt{3} \right)\)
83. Roots: \(1, \, -\frac{3}{2}, \, \frac{1}{2}\); \(4(x – 1) \left( x + \frac{3}{2} \right) \left( x – \frac{1}{2} \right)\)
84. Roots: \(-2, \, \frac{3}{2} + \frac{\sqrt{7}}{2}, \, \frac{3}{2} – \frac{\sqrt{7}}{2}\); \(2 (x + 2) \left( x – \frac{3}{2} + \frac{\sqrt{7}}{2} \right) \left( x – \frac{3}{2} – \frac{\sqrt{7}}{2} \right)\)
85. Roots: \(-1, \, 2, \, \sqrt{3}, \, -\sqrt{3}\); \((x + 1)(x – 2) \left( x – \sqrt{3} \right) \left( x + \sqrt{3} \right)\)
86. Roots: \(1, \, -1, \, -2, \, \frac{1}{3}\); \(3(x – 1)(x + 1)(x + 2) \left( x – \frac{1}{3} \right)\)
87. Roots: \(-1, \, -2, \, 1, \, 2, \, 3\); \((x + 1)(x+2)(x-1)(x-2)(x-3)\)
88. Roots: \(-1, \, -2, \, -3, \, 3\); \((x + 1)^2 (x + 2) (x + 3)(x – 3)\)

In exercises 1 through 14, solve the equations.

1. \[ 5(x – 3) = 3(x + 7) + x \]
2. \[ y – (6 – 2y) = 8(y – 2) \]
3. \[ \frac{1}{2} (2x-1) = 3\left(x+\frac{1}{4} \right) \]
4. \[ \frac{x}{4} – \frac{x}{3} = \frac{7}{6} – \frac{4x}{3} \]
5. \[ \frac{2x-1}{5} = \frac{2+x}{3} \]
6. \[ \frac{7z+1}{6} + \frac{3}{2} = \frac{3z}{4} \]
7. \[ \frac{x-1}{3} – \frac{2-3x}{14} = \frac{4x-3}{7} \]
8. \[ \frac{x-3}{6} – \frac{2x-1}{5} =-1 \]
9. \[ \frac{x+1}{5} + \frac{x+2}{6} = \frac{x-1}{4} + \frac{x+7}{10} \]
   \[ \begin{aligned} &\frac{x+1}{5} + \frac{x+2}{6} \\[1em] &\hspace{3em}= \frac{x-1}{4} + \frac{x+7}{10} \end{aligned} \]
10. \[ \frac{5x-2}{3} – \frac{1}{2} (3x-1) = \frac{9x+7}{6} – \frac{2}{9} (5x-1) \]
    \[ \begin{aligned} &\frac{5x-2}{3} – \frac{1}{2} (3x-1) \\[1em] &\hspace{3em}= \frac{9x+7}{6} – \frac{2}{9} (5x-1) \end{aligned} \]
11. \[ (x-3)^2 = (x-1)^2 \]
12. \[ (x-5)(x+1) = (x+2)(x-3) + 13 \]
    \[ \begin{aligned} &(x-5)(x+1) \\[1em] &\hspace{4em} = (x+2)(x-3) + 13 \end{aligned} \]
13. \[ (2x-5) (x-1) + x^2 = (3x-1) (x+2) +1 \]
    \[ \begin{aligned} &(2x-5) (x-1) + x^2 \\[1em] &\hspace{3em} = (3x-1) (x+2) +1 \end{aligned} \]
14. \[ 8x(x+2) (x-1) = (2x+1)^3 – (2x+3)^2 \]
    \[ \begin{aligned} &8x(x+2) (x-1) \\[1em] &\hspace{3em} = (2x+1)^3 – (2x+3)^2 \end{aligned} \]

In exercises 15 through 22, solve the equations for the variable \(\boldsymbol{x}\).

15. \[ 5(5x-a) = a^2 (x-1) \]
16. \[ a(x+b) + x(b-a) = 2b (2a-x) \]
17. \[ x^2 + b^2 + b(b-1) = (x+b)^2 \]
18. \[ (x+a)^3 -2x^3 = 12a^3 – (x-a) \]
19. \[ \frac{x-a}{b} + \frac{x-b}{a} = 2 \]
20. \[ \frac{x-3m}{m^2} + \frac{x-2m}{mn} = -\frac{1}{m} \]
21. \[ \frac{a-x}{a} – \frac{b-x}{b} = \frac{2(a-b)}{ab} \]
22. \[ \frac{x-a}{a+b} + \frac{a+b}{a-b} = \frac{x+b}{a+b} + \frac{x-b}{a-b} \]

In exercises 23 through 26, solve each equation for the indicated variable in terms of the remaining variables.

23. \[ A = \pi (r^2+rs) ,\; s \]
24. \[ S = a\frac{1-r^n}{1-r} ,\; a \]
25. \[ S = \frac{f}{H-h} ,\; h \]
26. \[ \frac{1}{x} + \frac{1}{y} = \frac{1}{a} ,\; x \]

Solve the equations from 27 to 40 using factorization.

27. \[ x^2-4x-12=0 \]
28. \[ x^2-6x+9=0 \]
29. \[ x^2+24=-11x \]
30. \[ 2x^2-3x+1=0 \]
31. \[ 9x^2-17x-2=0 \]
32. \[ (2x-1)^2 – (x+5)^2 =-19 \]
33. \[ (x-5)^2 – (x-4)^2 = (2x+3)^2 +12 \]
    \[ \begin{aligned} &(x-5)^2 – (x-4)^2 \\[1em] &\hspace{5em}= (2x+3)^2 +12 \end{aligned} \]
34. \[ (x-2)^3 – (x+1)^3 = -x(3x+4) -24 \]
    \[ \begin{aligned} &(x-2)^3 – (x+1)^3 \\[1em] &\hspace{4em}= -x(3x+4) -24 \end{aligned} \]
35. \[ 6x^2 -\frac{5x}{2} = -\frac{1}{4} \]
36. \[ \frac{2(x+5)}{5} + \frac{x-4}{4} = \frac{x^2-53}{5} \]
37. \[ x^4-17x^2+16=0 \]
38. \[ 6y^4 = \frac{y^2}{2} + \frac{1}{4} \]
39. \[ x^{2/3}+x^{1/3}-6=0 \]
40. \[ 2x^{2/3}+3x^{1/3}-2=0 \]

Solve the equations from 41 to 46 using the quadratic formula.

41. \[ 9(x-1)^2=5 \]
42. \[ \sqrt{3}x-3=4x^2 \]
43. \[ 2x(2x-3)=-1 \]
44. \[ (x+15)^2=6x(x+5) \]
45. \[ x^2-2x-(a^2+2a)=0 \]
46. \[ \frac{x^2}{2a} – \frac{a+2}{2a} x+1=0 \]

In exercises 47 through 60, solve the given equations.

47. \[ \frac{x-6}{x} = \frac{x+6}{x-6} + \frac{6}{x} \]
48. \[ \frac{x}{x+2} – \frac{x}{x-2} = \frac{x-15}{x^2-4} \]
49. \[ \frac{1}{3x-3} + \frac{1}{4x+4} = \frac{1}{12x-12} \]
50. \[ \frac{4x+1}{4x-1} = \frac{4x-1}{4x+1} + \frac{6}{16x^2-1} \]
51. \[ \frac{1}{x} + \frac{1}{4-x} = 1 \]
52. \[ \frac{x}{1+x} + \frac{1}{1-x} = 0 \]
53. \[ \frac{3y-2}{3y+2} = \frac{2y+3}{4y-1} \]
54. \[ \frac{ x+5 }{ (x-1)(x+2) } = \frac{2x}{x+2} \]
55. \[ \frac{1}{x-1} – \frac{1}{x-2} = \frac{1}{x-3} \]
56. \[ \frac{3x}{x-2} – \frac{1}{x^2-4} = 2 \]
57. \[ \frac{1}{x^2} + \frac{2}{x} -15 = 0 \]
58. \[ \frac{12}{x-1} + \frac{12}{x} = 10 \]
59. \[ \frac{2x}{x-1} = \frac{8}{x-1} – \frac{5}{x} \]
60. \[ \frac{1}{x^2-4} + \frac{2x+3}{x+2} + \frac{x+3}{x-2} = 0 \]

In exercises 61 through 76, solve the equations and eliminate extraneous solutions.

61. \[ 5-\sqrt{2x+3}= \]
62. \[ \sqrt{ \frac{x}{18} + 1 } = \frac{2}{3} \]
63. \[ (5x-1)^{1/2}=7 \]
64. \[ (y+9)^{3/2} = 4^3 \]
65. \[ \sqrt{x^2-5} = 5-x \]
66. \[ \sqrt{z+7} – \sqrt{z} = 1 \]
67. \[ \sqrt{ 9x^2 – 10x } = 3x-2 \]
68. \[ \sqrt{ \frac{1}{x} } – \sqrt{ \frac{8}{4x+1} } = 0 \]
69. \[ \sqrt{4x+1} + 1 = 2x \]
70. \[ \sqrt{x^2+5} = 2x-1 \]
71. \[ \sqrt{x+5} = 2\sqrt{x} – 1 \]
72. \[ \sqrt{x} + \sqrt{x-3} = \sqrt{x+5} \]
73. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
74. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
75. \[ \sqrt{ x + \sqrt{x+8} } = 2\sqrt{x} \]
76. \[ \frac{x}{2} = \frac{ \sqrt{x+2} – \sqrt{x-2} }{ \sqrt{x+2} + \sqrt{x-2} } \]

In the exercises 77 and 78, solve the equations by using a change of variable.

77. \[ \left( \frac{3x}{x+1} \right)^2 – \frac{6x}{x+1} = 8 \]
78. \[ \sqrt[3]{ \frac{5x+4}{x-1} } + \sqrt[3]{ \frac{x-1}{5x+4} } = \frac{5}{2} \]

In the exercises 79 and 80, use the remainder theorem to find the remainders.

79. \[ 3x^4 – 5x^3 – 4x^2 + 3x – 2, \; \text{ entre } (x-2) \]
    \[ \begin{aligned} &3x^4 – 5x^3 – 4x^2 + 3x – 2, \\[1em] &\hspace{4em}\text{entre } (x-2) \end{aligned} \]
80. \[ x^3 – 6x^2 +11x – 6, \; \text{ entre } (x+2) \]
    \[ \begin{aligned} &x^3 – 6x^2 +11x – 6, \\[1em] &\hspace{4em} \text{entre } (x+2) \end{aligned} \]

In exercises 81 through 88, find the roots of the equations and factorize the polynomials.

81. \[ x^3 + 2x^2 – x – 2 = 0 \]
82. \[ x^3 – 3x^2 + 2 = 0 \]
83. \[ 4x^3 – 7x^2 + 3 = 0 \]
84. \[ 2x^3 – 2x^2 – 11x + 2 = 0 \]
85. \[ x^4 – x^3 – 5x^2 + 3x + 6 = 0 \]
86. \[ 3x^4 + 5x^3 – 5x^2 – 5x + 2 = 0 \]
87. \[ x^5 -3x^4 -5x^3 +15x^2+4x -12 = 0 \]
    \[ \begin{aligned} x^5 -3x^4 -5x^3 &+15x^2 \\[1em] &+ 4x -12 = 0 \end{aligned} \]
88. \[ x^5 + 4x^4 – 4x^3 -34x^2 -45x-18 = 0 \]
    \[ \begin{aligned} x^5 + 4x^4 – 4x^3 &-34x^2 \\[1em] &-45x-18 = 0 \end{aligned} \]

In the exercises 89, 90 and 91, use the factor theorem to prove:

89. \(x – a\) is a factor of \(x^n – a^n\), where \(n\) is positive integer.
90. \(x + a\) is a factor of \(x^n – a^n\), where \(n\) is even integer.
91. \(x + a\) is a factor of \(x^n + a^n\), where \(n\) is odd integer.