1.6e: Exercises - Quadratic in Form

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A: Quadratic in Form Polynomial Equations

Exercise \(\PageIndex{A} \)

\(\bigstar \) Solve quadratic in form 4th degree equations.

\(x^{4}+x^{2}-72=0 \\[4pt] \)
\(x^{4}-17 x^{2}-18=0 \\[4pt] \)
\(x^{4}-13 x^{2}+36=0 \\[4pt] \)
\(4 x^{4}-17 x^{2}+4=0 \\[4pt] \)
\(y^{4}-14 y^{2}+46=0 \\[4pt] \)
\(x^{4}-6 x^{2}+6=0 \\[4pt] \)
\(x^{4}-6 x^{2}+7=0 \\[4pt] \)
\(x^{4}-12 x^{2}+31=0 \)

\(x^{4}-8 x^{2}+14=0 \\[4pt] \)
\(9 x^{4}-30 x^{2}+1=0 \\[4pt] \)
\(4 x^{4}-16 x^{2}+13=0 \\[4pt] \)
\(x^{4}-9 x^{2}+18=0 \\[4pt] \)
\(x^{4}-7 x^{2}+12=0 \\[4pt] \)
\(x^{4}+5 x^{2}-36=0 \\[4pt] \)
\(x^{4}-13 x^{2}-30=0 \\[4pt] \)
\(4 x^{4}-5 x^{2}+1=0 \)

\(2 x^{4}-5 x^{2}+3=0 \\[4pt] \)
\(3 x^{4}-14 x^{2}+8=0 \\[4pt] \)
\(2 x^{4}-7 x^{2}+3=0 \\[4pt] \)
\(x^{4}-2x^{2}-35=0 \\[4pt] \)
\(x^{4}-11x^{2}+18=0 \\[4pt] \)
\(6x^{4}-23x^{2}-225=0 \\[4pt] \)
\(12x^{4}-31x^{2}+9=0 \\[4pt] \)
\(x^{4}+19x^{2}+48=0 \)

Answers to odd exercises:

1. \(\pm 2 \sqrt{2}, \pm 3 i \)
3. \(\pm 2, \pm 3 \)
5. \(\pm \sqrt{7-\sqrt{3}}, \pm \sqrt{7+\sqrt{3}} \)
7. \(\pm 1.26, \pm 2.10 \) or \( \pm \sqrt {3 \pm \sqrt{2}} \)

9. \(\pm 1.61, \pm 2.33 \) or \( \pm \sqrt {4 \pm \sqrt{2}} \)
11. \(\pm 1.06, \pm 1.69 \) or \( \pm \sqrt {2 \pm \frac{1}{2}\sqrt{3}} \)
13. \(x=\pm \sqrt{3}, x=\pm 2 \)
15. \(x=\pm \sqrt{15}, x=\pm \sqrt{2} i \)

17. \(x=\pm 1, x=\frac{ \pm \sqrt{6}}{2} \)
19. \(x=\pm \sqrt{3}, x=\pm \frac{\sqrt{2}}{2} \)
21. \( \{ \pm 3, \pm \sqrt{2} \} \)
23. \( \{ \pm \frac {3}{2}. \pm \frac{\sqrt3}{3} \} \)

\(\bigstar \) Solve quadratic in form polynomial equations.

\(x^{6}+7 x^{3}-8=0 \\[4pt] \)
\(x^{6}+28 x^{3}+27=0 \\[4pt] \)
\(81 y^{4}-1=0 \\[4pt] \)
\(x^{6}+16 x^{3}+64=0 \)

\((x-3)^{2}-5(x-3)-36=0 \\[4pt] \)
\((x+2)^{2}-3(x+2)-54=0 \\[4pt] \)
\((3 y+2)^{2}+(3 y+2)-6=0 \\[4pt] \)
\((5 y-1)^{2}+3(5 y-1)-28=0 \)

\( \left(x^{2}+1\right)^{2} - 5\left(x^{2}+1\right)+4=0 \\[4pt] \)
\(\left(x^{2}-4\right)^{2}-4\left(x^{2}-4\right)+3=0 \\[4pt] \)
\(2\left(x^{2}-5\right)^{2}-5\left(x^{2}-5\right)+2=0 \\[4pt] \)
\(2\left(x^{2}-5\right)^{2}-7\left(x^{2}-5\right)+6=0 \)

43. \( (x^2-2x+3)^2 -5(x^2-2x+3) + 6 = 0 \)

44. \( (x^2+x-2)^2 -5(x^2+x-2) - 50 = 0 \)

Answers to odd exercises

31. \(-2,1,1 \pm i \sqrt{3},-\frac{1}{2} \pm \frac{\sqrt{3}}{2} i \)
33. \(\pm \frac{1}{3}, \pm \frac{i}{3} \)
35. \(x=12\), \(x=-1\)

37. \(x=-\frac{5}{3}, x=0 \)
39. \(x=0, x=\pm \sqrt{3} \)

41. \(x=\pm \sqrt{7}, x=\pm \frac{\sqrt{22}}{2} \)
43. \( \{ 0, 1, 1, 2 \} \)

B. Quadratic in Form Radical and Exponential Equations

Exercise \(\PageIndex{B} \)

\(\bigstar \) Solve quadratic in form radical equations.

\(x+2 \sqrt{x}-3=0 \\[4pt] \)
\(x-\sqrt{x}-2=0 \\[4pt] \)
\(x-5 \sqrt{x}+6=0 \\[4pt] \)
\(x-6 \sqrt{x}+5=0 \\[4pt] \)
\(x-\sqrt{x}-20=0 \\[4pt] \)
\(x-8 \sqrt{x}+15=0 \\[4pt] \)
\(x+6 \sqrt{x}-16=0 \)

\(x+4 \sqrt{x}-21=0 \\[4pt] \)
\(6 x+\sqrt{x}-2=0 \\[4pt] \)
\(6 x+\sqrt{x}-1=0 \\[4pt] \)
\(10 x-17 \sqrt{x}+3=0 \\[4pt] \)
\(12 x+5 \sqrt{x}-3=0 \\[4pt] \)
\(x+2\sqrt{x}-15=0 \\[4pt] \)
\(x-10\sqrt{x}+21=0 \)

\(x-8 x^{\frac{1}{2}}+7=0 \\[4pt] \)
\(2 x-7 x^{\frac{1}{2}}=15 \\[4pt] \)
\(x^{1 / 2}-3 x^{1 / 4}+2=0 \\[4pt] \)
\(x+5 \sqrt{x}-50=0 \\[4pt] \)
\(6x^{1 / 4}- x^{1 / 2}-9=0 \\[4pt] \)
\(x^{1 / 3}+ x^{1 / 6}-6=0 \\[4pt] \)
\(2 x^{1 / 3}-3 x^{1 / 6}+1=0 \\[4pt] \)
\(x^{1 / 3}-x^{1 / 6}-2=0 \)

Answers to odd exercises:

51. \(1 \)
53. \(4,9 \)
55. \(x=25 \)

57. \(x=4 \)
59. \(x=\frac{1}{4} \)

61. \(x=\frac{1}{25}, x=\frac{9}{4} \)
63. \( \{ 9 \} \)

65. \(x=1, x=49 \)
67. \(1, 16 \)

69. \( \{ 81 \} \)
71. \(\frac{1}{64}, 1 \)

\(\bigstar \) Solve quadratic in form radical equations.

\(x^{2 / 3}+5 x^{1 / 3}+6=0 \\[4pt] \)
\(x^{2 / 3}-2 x^{1 / 3}-35=0 \\[4pt] \)
\(4 x^{2 / 3}-4 x^{1 / 3}+1=0 \\[4pt] \)
\(3 x^{2 / 3}-2 x^{1 / 3}-1=0 \\[4pt] \)
\(8 x^{2 / 3}+7 x^{1 / 3}-1=0 \\[4pt] \)
\(x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28 \\[4pt] \)
\(x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12 \\[4pt] \)
\(x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0 \)

\(6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12 \\[4pt] \)
\(3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8 \\[4pt] \)
\(8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0 \\[4pt] \)
\(20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0 \\[4pt] \)
\(x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0 \\[4pt] \)
\(x^{2 / 3}+12 x^{1 / 3}+35=0 \\[4pt] \)
\(x^{2 / 3}-3 x^{1 / 3}-18=0 \\[4pt] \)
\(x^{2 / 3}- x^{1 / 3}+4=0 \)

\(x^{2 / 3}-2 x^{1 / 3}-6=0 \\[4pt] \)
\(9x^{2 / 3}-33 x^{1 / 3}+28=0 \\[4pt] \)
\(6x^{2 / 3}+11 x^{1 / 3}-10=0 \\[4pt] \)
\(y^{2 / 3}-9=0 \\[4pt] \)
\(30 y^{2 / 3}-15 y^{1 / 3}=0 \\[4pt] \)
\(x^{4 / 3}-2 x^{2 / 3}+1=0 \\[4pt] \)
\(x^{4 / 3}-13 x^{2 / 3}+36=0 \\[4pt] \)
\(x^{4 / 3}-5 x^{2 / 3}-6=0 \)

Answers to odd exercises:

81. \(-27, -8 \);
83. \(\frac{1}{8} \)
85. \(-1, \frac{1}{512} \)

87. \(x=8, x=-216 \)
89. \(x=\frac{27}{8}, x=-\frac{64}{27} \)
91. \(x=\frac{27}{512}, x=125 \)

93. \(x=-1, x=-512 \)
95. \( \{ 216, -27 \} \)
97. \( \{ 22 \pm 10\sqrt {7} \} \)

99. \( \{ \frac {8}{27},-\frac{125}{8} \} \)
101. \(0, \frac {1}{8} \)
103. \( \{ \pm 8, \pm 27 \} \)

C: Quadratic in Form Negative Exponential Equations

Exercise \(\PageIndex{C} \)

\(\bigstar \) Solve quadratic in form negative exponential equations.

\(5 x^{-2}+9 x^{-1}-2=0 \\[4pt] \)
\(3 x^{-2}+8 x^{-1}-3=0 \\[4pt] \)
\(8 x^{-2}+14 x^{-1}-15=0 \\[4pt] \)
\(9 x^{-2}-24 x^{-1}+16=0 \\[4pt] \)
\(2 y^{-2}-y^{-1}-1=0 \)

\(10 x^{-2}-19 x^{-1}-2=0 \\[4pt]\)
\(6 x^{-2}+13 x^{-1}+5=0 \\[4pt] \)
\(15 x^{-2}-26 x^{-1}+8=0 \\[4pt] \)
\(8 x^{-2}-2 x^{-1}-3=0 \\[4pt] \)
\(15 x^{-2}-4 x^{-1}-4=0 \)

\(4 y^{-2}-9=0 \\[4pt] \)
\(16 y^{-2}+4 y^{-1}=0 \\[4pt] \)
\(2 x^{-2 / 3}-3 x^{-1 / 3}-2=0 \\[4pt] \)
\(4 x^{-1}-17 x^{-1 / 2}+4=0 \\[4pt] \)
\(3 x^{-1}-8 x^{-1 / 2}+4=0 \)

Answers to odd exercises:

111. \(-\frac{1}{2}, 5 \)
113. \(-\frac{2}{5}, \frac{4}{3} \)

115. \(-2,1 \)
117. \(x=-10, x=\frac{1}{2} \)

119. \(x=\frac{3}{4}, x=\frac{5}{2} \).
121. \( x=\frac{3}{2}, x=-\frac{5}{2} \)

125. \(y = -4\)
127. \(\frac{1}{16}, 16 \)

D: Quadratic in Form Rational Equations

Exercise \(\PageIndex{D} \): Quadratic in Form Rational Equations

\(\bigstar \) Solve quadratic in form rational equations.

17. \(\left(\dfrac{x-3}{x}\right)^{2}-2\left(\dfrac{x-3}{x}\right)-24=0 \\[4pt] \)
\(\left(\dfrac{2 x+1}{x}\right)^{2}+9\left(\dfrac{2 x+1}{x}\right)-36=0 \\[4pt] \)
\(2\left(\dfrac{x}{x+1}\right)^{2}-5\left(\dfrac{x}{x+1}\right)-3=0 \\[4pt] \)
\(3\left(\dfrac{x}{3 x-1}\right)^{2}+13\left(\dfrac{x}{3 x-1}\right)-10=0 \\[4pt] \)
\(12\left(\dfrac{x}{2 x-3}\right)^{2}-11\left(\dfrac{x}{2 x-3}\right)+2=0 \\[4pt] \)
\(5\left(\dfrac{1}{x+2}\right)^{2}-3\left(\dfrac{1}{x+2}\right)-2=0 \\[4pt] \)

\(\left( \dfrac{2x}{x-5} \right)^{2} +\dfrac{6x}{x-5}-18=0 \\[4pt] \)
\( \dfrac{x^2+2}{x-6}+\dfrac{x-6}{x^2+2} +\dfrac{13}{6}=0 \)
\( \dfrac{x^2+2}{x^2-2}+\dfrac{x^2-2}{x^2+2} -\dfrac{170}{77}=0 \\[4pt] \)
\( \sqrt{\dfrac{x+10}{x-6}} + \sqrt{\dfrac{x-6}{x+10} }-\dfrac{34}{15}=0 \\[4pt] \)
\( \sqrt{\dfrac{x+7}{x-5}} - \sqrt{\dfrac{x-5}{x+7} }-\dfrac{3}{2} =0 \)

Answers to odd exercises:

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

133. \(-\frac{3}{2},-\frac{1}{3} \)

135. \(-\frac{3}{2}, 6 \)

137. \( \{ 15, \frac{15}{4} \} \)

139. \( \{ \pm 3, \; \pm 3i \} \)

141. \( \{ 9 \} \)

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