Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.2e: Exercises - SqRP, CTS, QF

  • Page ID
    45461
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    A: Square Root Property

    Exercise \(\PageIndex{A}\)

    \( \bigstar \) Solve each equation by the square root property

    1. \(x ^ { 2 } = 81 \\[4pt] \)

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

    3. \(y ^ { 2 } = \dfrac { 1 } { 9 } \\[4pt] \)

    4. \(y ^ { 2 } = \dfrac { 1 } { 16 } \\[4pt] \)

    5. \(x ^ { 2 } = 12 \\[4pt] \)

    6. \(x ^ { 2 } = 18 \\[4pt] \)

    7. \(16 x ^ { 2 } = 9 \\[4pt] \)

    8. \(4 x ^ { 2 } = 25 \\[4pt] \)

    9. \(2 t ^ { 2 } = 1 \\[4pt] \)

    10. \(3 t ^ { 2 } = 2 \\[4pt] \)

    11. \(x ^ { 2 } - 16=0 \\[4pt] \)

    12. \(x ^ { 2 } - 36=0 \\[4pt] \)

    13. \(x ^ { 2 } - 40 = 0 \\[4pt] \)

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

    15. \(x ^ { 2 } + 1 = 0 \\[4pt] \)

    16. \(x ^ { 2 } + 100 = 0 \\[4pt] \)

    Answers to Odd Exercises: 

    1. \(\pm 9  \)

    3. \(\pm \frac{1}{3}  \)

    5. \(\pm 2 \sqrt { 3 }  \)

    7. \(\pm \frac { 3 } { 4 }  \)

    9. \(\pm \frac { \sqrt { 2 } } { 2 } \)

    11. \(-4,4  \)

    13. \(\pm 2 \sqrt { 10 } \)

    15. \(\pm i  \)

    \( \bigstar \) Solve each equation by the square root property

    21. \(9 y ^ { 2 } - 1 = 0 \\[4pt] \)

    22. \(4 y ^ { 2 } - 25 = 0 \\[4pt] \)

    23. \(5 x ^ { 2 } - 1 = 0 \\[4pt] \)

    24. \(6 x ^ { 2 } - 5 = 0 \\[4pt] \)

    25. \(8 x ^ { 2 } + 1 = 0 \\[4pt] \)

    26. \(12 x ^ { 2 } + 5 = 0 \\[4pt] \)

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

    28. \(x ^ { 2 } - \dfrac { 9 } { 25 } = 0 \\[4pt] \)

    29. \(y ^ { 2 } + 6 = 2 \\[4pt] \)

    30. \(y ^ { 2 } + 8 = 7 \\[4pt] \)

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

    32. \(t ^ { 2 } - 14 = 4 \\[4pt] \)

    33. \(3x ^ { 2 } + 25 = 1 \\[4pt] \)

    34. \(2x ^ { 2 } + 81 = 31 \\[4pt] \)

    35. \(5 y ^ { 2 } +7 = 9 \\[4pt] \)

    36. \(3 x ^ { 2 } +4 = 5 \\[4pt] \)

    Answers to Odd Exercises: 

    21. \(- \frac { 1 } { 3 } , \frac { 1 } { 3 }  \)

    23. \(\pm \frac { \sqrt { 5 } } { 5 } \)

    25. \(\pm \frac { \sqrt { 2 } } { 4 } i \)

    27. \(\pm \frac { 2 } { 3 } \)

    29. \(\pm 2 i  \)

    31. \(\pm 2 \sqrt { 2 } \)

    33. \(\pm 2 i \sqrt { 2 } \)

    35. \(\pm \frac { \sqrt { 10 } } { 5 }  \)

    \( \bigstar\) Solve each equation by the square root property

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

    42. \(( x + 1 ) ^ { 2 } - 4 = 0 \\[4pt] \)

    43. \(( u - 5 ) ^ { 2 } - 25 = 0 \\[4pt] \)

    44. \(( u + 2 ) ^ { 2 } - 4 = 0 \\[4pt] \)

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

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

    47. \(( x - 5 ) ^ { 2 } - 20 = 0 \\[4pt] \)

    48. \(( x + 1 ) ^ { 2 } - 28 = 0 \\[4pt] \)

    49. \(( 3 t + 2 ) ^ { 2 } + 6 = 0 \\[4pt] \)

    50. \(( 3 t - 5 ) ^ { 2 } + 10 = 0 \\[4pt] \)

    51. \(4 ( y - 2 ) ^ { 2 } - 9 = 0 \\[4pt] \)

    52. \(9 ( y + 1 ) ^ { 2 } - 4 = 0 \\[4pt] \)

    53. \(4 ( 3 x + 1 ) ^ { 2 } - 27 = 0 \\[4pt] \)

    54. \(9 ( 2 x - 3 ) ^ { 2 } - 8 = 0 \\[4pt] \)

    55. \(2 ( 3 x - 1 ) ^ { 2 } + 3 = 0 \\[4pt] \)

    56. \(5 ( 2 x - 1 ) ^ { 2 } + 2 = 0 \\[4pt] \)

    57. \(3 \left( y - \dfrac { 2 } { 3 } \right) ^ { 2 } - \dfrac { 3 } { 2 } = 0 \\[4pt] \)

    58. \(2 \left( 3 y - \dfrac { 1 } { 3 } \right) ^ { 2 } - \dfrac { 5 } { 2 } = 0 \\[4pt] \)

    59. \(- 3 ( t - 1 ) ^ { 2 } + 12 = 0 \\[4pt] \)

    60. \(- 2 ( t + 1 ) ^ { 2 } + 8 = 0 \\[4pt] \)

    Answers to Odd Exercises: 

    41. \(1, 3 \)

    43. \(0,10\)

    45. \(-9,-5 \)

    47. \(5 \pm 2 \sqrt { 5 }  \)

    49. \(- \frac { 2 } { 3 } \pm \frac { \sqrt { 6 } } { 3 } i \)

    51. \(\frac { 1 } { 2 } , \frac { 7 } { 2 } \)

    53. \(\frac { - 2 \pm 3 \sqrt { 3 } } { 6 } \)

    55. \(\frac { 1 } { 3 } \pm \frac { \sqrt { 6 } } { 6 } i  \)

    57. \(\frac { 4\pm 3 \sqrt { 2 } } { 6 }  \)

    59. \(-1,3 \)

    B: Complete the Square

    Exercise \(\PageIndex{B}\)

    \( \bigstar \) Determine the constant that should be added to the binomial and then complete the square

    1. \(x ^ { 2 } - 2 x + ? = ( x - ? ) ^ { 2 } \\[4pt] \)
    2. \(x ^ { 2 } - 4 x + ? = ( x - ? ) ^ { 2 } \\[4pt] \)
    3. \(x ^ { 2 } + 10 x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    1. \(x ^ { 2 } + 12 x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    2. \(x ^ { 2 } + 7 x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    3. \(x ^ { 2 } + 5 x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    1. \(x ^ { 2 } - x + ? = ( x - ? ) ^ { 2 } \\[4pt] \)
    2. \(x ^ { 2 } - \dfrac { 1 } { 2 } x + ? = ( x - ? ) ^ { 2 } \\[4pt] \)
    1. \(x ^ { 2 } + \dfrac { 2 } { 3 } x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    2. \(x ^ { 2 } + \dfrac { 4 } { 5 } x + ? = ( x + ? ) ^ { 2 } \\[4pt] \)
    Answers to Odd Exercises:
    61. \(x ^ { 2 } - 2 x + 1 = ( x - 1 ) ^ { 2 }  \)
    63. \(x ^ { 2 } + 10 x + 25 = ( x + 5 ) ^ { 2 }  \)
    65. \(x ^ { 2 } + 7 x + \frac { 49 } { 4 } = \left( x + \frac { 7 } { 2 } \right) ^ { 2 }  \)
    67. \(x ^ { 2 } - x + \frac { 1 } { 4 } = \left( x - \frac { 1 } { 2 } \right) ^ { 2 } \)
    69. \(x ^ { 2 } + \frac { 2 } { 3 } x + \frac { 1 } { 9 } = \left( x + \frac { 1 } { 3 } \right) ^ { 2 } \)

    \( \bigstar \) Solve each equation by completing the square

    71. \(x ^ { 2 } + 2 x = 8 \\[4pt] \)

    72. \(x ^ { 2 } - 8 x =- 15 \\[4pt] \)

    73. \(y ^ { 2 } + 2 y = 24 \\[4pt] \)

    74. \(y ^ { 2 } - 12 y =- 11 \\[4pt] \)

    75. \(x^{2}-4 x-1=15 \\[4pt] \)

    76. \(x^{2}-12 x+8=-10 \\[4pt] \)

    77. \(x(x+1)-11(x-2)=0 \\[4pt] \)

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

    79. \(2 y ^ { 2 } - y - 1 = 0 \\[4pt] \)

    80. \(2 y ^ { 2 } + 7 y - 4 = 0 \\[4pt] \)

    81. \(x^{2}+6 x-1=0 \\[4pt] \)

    82. \(x^{2}+8 x+10=0 \\[4pt] \) b

    83. \(x^{2}-2 x-7=0 \\[4pt] \)

    84. \(x^{2}-6 x-3=0 \\[4pt] \)

    85. \(y^{2}-2 y+4=0 \\[4pt] \)

    86. \(y^{2}-4 y+9=0 \\[4pt] \)

    87. \(t^{2}+10 t-75=0 \\[4pt] \)

    88. \(t^{2}+12 t-108=0 \\[4pt] \)

    Answers to Odd Exercises:

    71. \(-4,2 \)

    73. \(-6,4\)

    75. 2\(\pm 2 \sqrt{5}  \)

    77. 5\(\pm \sqrt{3}  \)

    79. \(- \frac { 1 } { 2 } , 1 \)

    81. \(-3 \pm \sqrt{10}  \)

    83. 1\(\pm 2 \sqrt{2} \)

    85. 1\(\pm i \sqrt{3}  \)

    87. \(-15,5  \)

    \( \bigstar \) Solve each equation by completing the square

    91. \(t ^ { 2 } + 3 t = 28 \\[4pt] \)

    92. \(t ^ { 2 } - 7 t =- 10 \\[4pt] \)

    93. \(x^{2}+x-1=0 \\[4pt] \)

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

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

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

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

    98. \(x^{2}+x+1=0 \\[4pt] \)

    99. \(y^{2}=(2 y+3)(y-1)-2(y-1) \\[4pt] \)

    100. \((2 y+5)(y-5)-y(y-8)=-24 \\[4pt] \)

    101. \(x^{2}-7 x+\dfrac{11}{2}=0 \\[4pt] \)

    102. \(x^{2}-9 x+\dfrac{3}{2}=0 \\[4pt] \)

    103. \(t^{2}-\dfrac{1}{2} t-1=0 \\[4pt] \)

    104. \(t^{2}-\dfrac{1}{3} t-2=0 \\[4pt] \)

    105. \(u^{2}-\dfrac{2}{3} u-\dfrac{1}{3}=0 \\[4pt] \)

    106. \(u^{2}-\dfrac{4}{5} u-\dfrac{1}{5}=0 \\[4pt] \)

     

    Answers to Odd Exercises:

    91. \(-7,4 \)

    93. \(\frac{-1 \pm \sqrt{5}}{2} \)

    95. \(\frac{-3 \pm \sqrt{17}}{2} \)

    97. \(-\frac{3}{2} \pm \frac{\sqrt{11}}{2} i  \)

    99. \(\frac{1 \pm \sqrt{5}}{2}  \)

    101. \(\frac{7 \pm 3 \sqrt{3}}{2}  \)

    103. \(\frac{1 \pm \sqrt{17}}{4}  \)

    105. \(-\frac{1}{3}, 1  \)

    \( \bigstar \) Solve each equation by completing the square

    1. \(2 x^{2}-4 x+10=0 \\[4pt] \)
    2. \(6 x^{2}-24 x+42=0 \\[4pt] \)
    3. \(4 x^{2}-8 x-1=0 \\[4pt] \)
    4. \(2 x^{2}-4 x-3=0 \\[4pt] \)
    5. \(3 x^{2}+6 x+1=0 \\[4pt] \)
    6. \(5 x^{2}+10 x+2=0 \\[4pt] \)
    7. \(4 x^{2}-12 x-15=0 \\[4pt] \)
    8. \(2 x ^ { 2 } + 3 x - 2 = 0 \\[4pt] \)
    9. \(3x^2-x-2=0 \\[4pt] \)
    1. \(2 x^{2}+4 x-43=0 \\[4pt] \)
    2. \(3 x^{2}+2 x-3=0 \\[4pt] \)
    3. \(5 x^{2}+2 x-5=0 \\[4pt] \)
    4. \(2 x^{2}-x-2=0 \\[4pt] \)
    5. \(2 x^{2}+3 x-1=0 \\[4pt] \)
    6. \(3 u^{2}+2 u+2=0 \\[4pt] \)
    7. \(3 u^{2}-u+1=0 \\[4pt] \)
    8. \(2 y ^ { 2 } - y - 1 = 0 \\[4pt] \)
    9. \(2 y ^ { 2 } + 7 y - 4 = 0 \\[4pt] \)
    1. \((t+2)^{2}=3(3 t+1) \\[4pt] \)
    2. \((3 t+2)(t-4)-(t-8)=1-10 t \\[4pt] \)
    3. \((2 x-1)^{2}=2 x \\[4pt] \)
    4. \((3 x-2)^{2}=5-15 x \\[4pt] \)
    5. \((2 x+1)(3 x+1)=9 x+4 \\[4pt] \)
    6. \((3 x+1)(4 x-1)=17 x-4 \\[4pt] \)
    7. \(9 x(x-1)-2(2 x-1)=-4 x \\[4pt] \)
    8. \((6 x+1)^{2}-6(6 x+1)=0 \\[4pt] \)
    Answers to Odd Exercises:

    111. 1\(\pm 2 i  \)

    113. \(\frac{2 \pm \sqrt{5}}{2} \)

    115. \(\frac{-3 \pm \sqrt{6}}{3}  \)

    117. \(\frac{3 \pm 2 \sqrt{6}}{2} \)

    119. \(1,-\frac{2}{3} \)

    121. \(\frac{-1 \pm \sqrt{10}}{3} \)

    123. \(\frac{1 \pm \sqrt{17}}{4}  \)

    125. \(\frac{-1 \pm i\sqrt{5}}{3} \)

    127. \(- \frac { 1 } { 2 } , 1  \)

    129. \(\frac{5 \pm \sqrt{21}}{2} \)

    131. \(\frac{3 \pm \sqrt{5}}{4}  \)

    133. \(\frac{2 \pm \sqrt{22}}{6} \)

    135. \( \frac{1}{3}, \frac{2}{3}  \)

     

    C: Quadratic Formula

    Exercise \(\PageIndex{C}\) 

    \( \bigstar \) Solve each equation using the Quadratic Formula

    1. \(x^{2}-6 x-16=0 \\[4pt] \)
    2. \(x^{2}-3 x-18=0 \\[4pt] \)
    3. \(2 x^{2}+7 x-4=0 \\[4pt] \)
    4. \(3 x^{2}+5 x-2=0 \\[4pt] \)
    1. \(-x^{2}+9 x-20=0 \\[4pt] \)
    2. \(-2 x^{2}-3 x+5=0 \\[4pt] \)
    3. \(16 y^{2}-24 y+9=0 \\[4pt] \)
    4. \(4 y^{2}-20 y+25=0 \\[4pt] \)
    1. \(x^{2}-5 x+1=0 \\[4pt] \)
    2. \(x^{2}-7 x+2=0 \\[4pt] \)
    3. \(x^{2}+8 x+5=0 \\[4pt] \)
    4. \(x^{2}-4 x+2=0 \\[4pt] \)
    5. \(5 u^{2}-2 u+1=0 \\[4pt] \)
    1. \(8 u^{2}-20 u+13=0 \\[4pt] \)
    2. \(-y^{2}+16 y-62=0 \\[4pt] \)
    3. \(-y^{2}+14 y-46=0 \\[4pt] \)
    4. \(-2 t^{2}+4 t+3=0 \\[4pt] \)
    5. \(-4 t^{2}+8 t+1=0 \\[4pt] \)
    Answers to Odd Exercises:

    141. \(-2,8 \\[4pt] \)

    143. \(-4, \frac{1}{2} \)

    145. \(4,5  \)

    147. \(\frac{3}{4}  \)

    149. \(\frac{5 \pm \sqrt{21}}{2}  \)

    151. \(-4\pm \sqrt{11}  \)

    153. \(\frac{1}{5} \pm \frac{2}{5} i \)

    155. \(8\pm \sqrt{2} \)

    157. \(\frac{2 \pm \sqrt{10}}{2} \)

    \( \bigstar \) Solve each equation using the Quadratic Formula

    1. \(4 y^{2}-9=0 \\[4pt] \)
    2. \(9 y^{2}-25=0 \\[4pt] \)
    3. \(5 t^{2}-6 t=0 \\[4pt] \)
    4. \(t^{2}+6 t=0 \\[4pt] \)
    1. \(x^{2}-18=0 \\[4pt] \)
    2. \(x^{2}-12=0 \\[4pt] \)
    3. \(x^{2}+12=0 \\[4pt] \)
    4. \(x^{2}+20=0 \\[4pt] \)
    1. \(3 x^{2}+2=0 \\[4pt] \)
    2. \(5 x^{2}+3=0 \\[4pt] \)
    3. \(y^{2}=2 y-10 \\[4pt] \)
    4. \(y^{2}=4 y-13 \\[4pt] \)
    1. \(2 x^{2}-10 x=1 \\[4pt] \)
    2. \(2 x^{2}-4 x=3 \\[4pt] \)
    3. \(3 x^{2}+2=x \\[4pt] \)
    4. \(4 x^{2}+1=3x \\[4pt] \)
    Answers to Odd Exercises:

    161. \(\pm \frac{3}{2} \) 

    163. \(0, \frac{6}{5}\)

    165. \(\pm 3 \sqrt{2} \)

    167. \(\pm 2 i \sqrt{3} \)

    169. \(\pm \frac{i \sqrt{6}}{3} \)

    171. \(1\pm 3 i  \)

    173. \(\frac{5 \pm 3 \sqrt{3}}{2} \)

    175. \(\frac{1}{6} \pm \frac{\sqrt{23}}{6} i  \)

    \( \bigstar  \) Solve each equation using the Quadratic Formula

    1. \(\dfrac{1}{2} y^{2}+5 y+\dfrac{3}{2}=0 \\[4pt] \)
    2. \(3 y^{2}+\dfrac{1}{2} y-\dfrac{1}{3}=0 \\[4pt] \)
    1. \(2 x^{2}-\dfrac{1}{2} x+\dfrac{1}{4}=0 \\[4pt] \)
    2. \(3 x^{2}-\dfrac{2}{3} x+\dfrac{1}{3}=0 \\[4pt] \)
    1. \(1.2 x^{2}-0.5 x-3.2=0 \\[4pt] \)
    2. \(0.4 x^{2}+2.3 x+1.1=0 \\[4pt] \)
    1. \(2.5 x^{2}-x+3.6=0 \\[4pt] \)
    2. \(-0.8 x^{2}+2.2 x-6.1=0 \\[4pt] \)
    Answers to Odd Exercises:

    181. \(-5 \pm \sqrt{22}  \)

    183. \(\frac{1}{8} \pm \frac{\sqrt{7}}{8} i  \)

    185. \(x \approx-1.4 \\[4pt] \) or \(x \approx 1.9 \)

    187. \(x \approx 0.2 \pm 1.2 i \)

    \( \bigstar \) Solve each equation using the Quadratic Formula

    1. \((x+2)^{2}+9=0 \\[4pt] \)
    2. \((x-4)^{2}+1=0 \\[4pt] \)
    3. \((2 x+1)^{2}-2=0 \\[4pt] \)
    4. \((3 x+1)^{2}-5=0 \\[4pt] \)
    1. \(-2 y^{2}=3(y-1) \\[4pt] \)
    2. \(3 y^{2}=5(2 y-1) \\[4pt] \)
    3. \((t+1)^{2}=2 t+7 \\[4pt] \)
    4. \((2 t-1)^{2}=73-4 t \\[4pt] \)
    1. \((x+5)(x-1)=2 x+1 \\[4pt] \)
    2. \((x+7)(x-2)=3(x+1) \\[4pt] \)
    3. \(2 x(x-1)=-1 \\[4pt] \)
    4. \(x(2 x+5)=3 x-5 \\[4pt] \)
    1. \(3 t(t-2)+4=0 \\[4pt] \)
    2. \(5 t(t-1)=t-4 \\[4pt] \)
    3. \((2 x+3)^{2}=16 x+4 \\[4pt] \)
    4. \((2 y+5)^{2}-12(y+1)=0 \\[4pt] \)
    Answers to Odd Exercises:

    191. \(-2 \pm 3 i  \)

    193. \(\frac{-1 \pm \sqrt{2}}{2}  \)

    195. \(\frac{-3 \pm \sqrt{33}}{4} \)

    197. \(\pm \sqrt{6}  \)

    199. \(-1 \pm \sqrt{7}\)

    201. \(\frac{1}{2} \pm \frac{1}{2} i  \)

    203. \(1\pm \frac{\sqrt{3}}{3} i  \)

    205. \(\frac{1}{2} \pm i  \)

    D: Factor and Solve (Sum and Difference of Cubes)

    Exercise \(\PageIndex{D}\)

    \( \bigstar \) Solve.

    1. \(x^3=8 \\[4pt] \)
    2. \(x^3=1 \\[4pt] \)
    1. \(64x^3=-27 \\[4pt] \)
    2. \(27x^3=-64 \\[4pt] \)
    1. \(8x^3=125 \\[4pt] \)
    2. \(125x^3 = 8 \\[4pt] \)
    1. \(2x^4 = 54x \\[4pt] \)
    2. \(3x^4 + 192x=0 \\[4pt] \)
    1. \(729x^6-1=0 \\[4pt] \)
    2. \(x^6=64 \\[4pt] \)
    Answers to Odd Exercises:

    211. \( \{ 2, -1 \pm i\sqrt{3} \}  \)

    213. \( \{ -\frac{3}{4}, \frac{3}{8} \pm \frac{3\sqrt{3}}{8}i \} \)

    215. \( \{ \frac{5}{2}, -\frac{5}{4} \pm \frac{5\sqrt{3}}{4}i \}  \)

    217. \( \{0, \; 3, \; -\frac{3}{2} \pm \frac{3\sqrt{3}}{2}i \} \)

    219. \( \{  \pm \frac{1}{3} ,  \frac{1}{6} \pm \frac{\sqrt{3}}{6}i, -\frac{1}{6} \pm \frac{\sqrt{3}}{6}i   \}  \)

    E: The Discriminant

    Exercise \(\PageIndex{E}\)

    \( \bigstar \) Calculate the discriminant and use it to determine the number and type of solutions. Do not solve.

    1. \(x^{2}-x+1=0 \\[4pt] \)
    2. \(x^{2}+2 x+3=0 \\[4pt] \)
    3. \(x^{2}-2 x-3=0 \\[4pt] \)
    4. \(x^{2}-5 x-5=0 \\[4pt] \)
    1. \(3 x^{2}-1 x-2=0 \\[4pt] \)
    2. \(3 x^{2}-1 x+2=0 \\[4pt] \)
    3. \(9 y^{2}+2=0 \\[4pt] \)
    4. \(9 y^{2}-2=0 \\[4pt] \)
    1. \(2 x^{2}+3 x=0 \\[4pt] \)
    2. \(4 x^{2}-5 x=0 \\[4pt] \)
    3. \(\dfrac{1}{2} x^{2}-2 x+\frac{5}{2}=0 \\[4pt] \)
    4. \(\dfrac{1}{2} x^{2}-x-\frac{1}{2}=0 \\[4pt] \)
    1. \(-x^{2}-3 x+4=0 \\[4pt] \)
    2. \(-x^{2}-5 x+3=0 \\[4pt] \)
    3. \(25 t^{2}+30 t+9=0 \\[4pt] \)
    4. \(9 t^{2}-12 t+4=0 \\[4pt] \)
    Answers to Odd Exercises:
    221. \(-3 \); two complex solutions
    223. \(16 \); two rational solutions
    225. \(-23 \); two complex solutions
    227. \(72\); two irrational solutions
    229. \(25 \); two rational solutions
    231. \(2 \); two irrational solutions
    233. \(37 \); two irrational solutions
    235. \(0 \); one rational solution

    \( \bigstar \\[4pt] \)


    1.2e: Exercises - SqRP, CTS, QF is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?