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.

 $$x^{4}+x^{2}-72=0 \\[4pt]$$ $$x^{4}-17 x^{2}-18=0 \\[4pt]$$ $$x^{4}-13 x^{2}+36=0 \\[4pt]$$ $$4 x^{4}-17 x^{2}+4=0 \\[4pt]$$ $$y^{4}-14 y^{2}+46=0 \\[4pt]$$ $$x^{4}-6 x^{2}+6=0 \\[4pt]$$ $$x^{4}-6 x^{2}+7=0 \\[4pt]$$ $$x^{4}-12 x^{2}+31=0$$ $$x^{4}-8 x^{2}+14=0 \\[4pt]$$ $$9 x^{4}-30 x^{2}+1=0 \\[4pt]$$ $$4 x^{4}-16 x^{2}+13=0 \\[4pt]$$ $$x^{4}-9 x^{2}+18=0 \\[4pt]$$ $$x^{4}-7 x^{2}+12=0 \\[4pt]$$ $$x^{4}+5 x^{2}-36=0 \\[4pt]$$ $$x^{4}-13 x^{2}-30=0 \\[4pt]$$ $$4 x^{4}-5 x^{2}+1=0$$ $$2 x^{4}-5 x^{2}+3=0 \\[4pt]$$ $$3 x^{4}-14 x^{2}+8=0 \\[4pt]$$ $$2 x^{4}-7 x^{2}+3=0 \\[4pt]$$ $$x^{4}-2x^{2}-35=0 \\[4pt]$$ $$x^{4}-11x^{2}+18=0 \\[4pt]$$  $$6x^{4}-23x^{2}-225=0 \\[4pt]$$  $$12x^{4}-31x^{2}+9=0 \\[4pt]$$ $$x^{4}+19x^{2}+48=0$$
 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. $$(x^2+x-2)^2 -5(x^2+x-2) - 50 = 0$$
 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. $$\{ 0, 1, 1, 2 \}$$

Exercise $$\PageIndex{B}$$

$$\bigstar$$ Solve quadratic in form radical equations.

 $$x+2 \sqrt{x}-3=0 \\[4pt]$$ $$x-\sqrt{x}-2=0 \\[4pt]$$ $$x-5 \sqrt{x}+6=0 \\[4pt]$$ $$x-6 \sqrt{x}+5=0 \\[4pt]$$ $$x-\sqrt{x}-20=0 \\[4pt]$$ $$x-8 \sqrt{x}+15=0 \\[4pt]$$ $$x+6 \sqrt{x}-16=0$$ $$x+4 \sqrt{x}-21=0 \\[4pt]$$ $$6 x+\sqrt{x}-2=0 \\[4pt]$$ $$6 x+\sqrt{x}-1=0 \\[4pt]$$ $$10 x-17 \sqrt{x}+3=0 \\[4pt]$$ $$12 x+5 \sqrt{x}-3=0 \\[4pt]$$ $$x+2\sqrt{x}-15=0 \\[4pt]$$ $$x-10\sqrt{x}+21=0$$ $$x-8 x^{\frac{1}{2}}+7=0 \\[4pt]$$ $$2 x-7 x^{\frac{1}{2}}=15 \\[4pt]$$ $$x^{1 / 2}-3 x^{1 / 4}+2=0 \\[4pt]$$ $$x+5 \sqrt{x}-50=0 \\[4pt]$$ $$6x^{1 / 4}- x^{1 / 2}-9=0 \\[4pt]$$ $$x^{1 / 3}+ x^{1 / 6}-6=0 \\[4pt]$$ $$2 x^{1 / 3}-3 x^{1 / 6}+1=0 \\[4pt]$$ $$x^{1 / 3}-x^{1 / 6}-2=0$$
 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.

 $$x^{2 / 3}+5 x^{1 / 3}+6=0 \\[4pt]$$ $$x^{2 / 3}-2 x^{1 / 3}-35=0 \\[4pt]$$ $$4 x^{2 / 3}-4 x^{1 / 3}+1=0 \\[4pt]$$ $$3 x^{2 / 3}-2 x^{1 / 3}-1=0 \\[4pt]$$ $$8 x^{2 / 3}+7 x^{1 / 3}-1=0 \\[4pt]$$  $$x^{\frac{2}{3}}-3 x^{\frac{1}{3}}=28 \\[4pt]$$ $$x^{\frac{2}{3}}+4 x^{\frac{1}{3}}=12 \\[4pt]$$ $$x^{\frac{2}{3}}-11 x^{\frac{1}{3}}+30=0$$ $$6 x^{\frac{2}{3}}-x^{\frac{1}{3}}=12 \\[4pt]$$ $$3 x^{\frac{2}{3}}-10 x^{\frac{1}{3}}=8 \\[4pt]$$ $$8 x^{\frac{2}{3}}-43 x^{\frac{1}{3}}+15=0 \\[4pt]$$ $$20 x^{\frac{2}{3}}-23 x^{\frac{1}{3}}+6=0 \\[4pt]$$ $$x^{\frac{2}{3}}+9 x^{\frac{1}{3}}+8=0 \\[4pt]$$ $$x^{2 / 3}+12 x^{1 / 3}+35=0 \\[4pt]$$ $$x^{2 / 3}-3 x^{1 / 3}-18=0 \\[4pt]$$ $$x^{2 / 3}- x^{1 / 3}+4=0$$ $$x^{2 / 3}-2 x^{1 / 3}-6=0 \\[4pt]$$ $$9x^{2 / 3}-33 x^{1 / 3}+28=0 \\[4pt]$$ $$6x^{2 / 3}+11 x^{1 / 3}-10=0 \\[4pt]$$ $$y^{2 / 3}-9=0 \\[4pt]$$ $$30 y^{2 / 3}-15 y^{1 / 3}=0 \\[4pt]$$ $$x^{4 / 3}-2 x^{2 / 3}+1=0 \\[4pt]$$ $$x^{4 / 3}-13 x^{2 / 3}+36=0 \\[4pt]$$ $$x^{4 / 3}-5 x^{2 / 3}-6=0$$
 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.

 $$5 x^{-2}+9 x^{-1}-2=0 \\[4pt]$$ $$3 x^{-2}+8 x^{-1}-3=0 \\[4pt]$$ $$8 x^{-2}+14 x^{-1}-15=0 \\[4pt]$$ $$9 x^{-2}-24 x^{-1}+16=0 \\[4pt]$$ $$2 y^{-2}-y^{-1}-1=0$$ $$10 x^{-2}-19 x^{-1}-2=0 \\[4pt]$$ $$6 x^{-2}+13 x^{-1}+5=0 \\[4pt]$$ $$15 x^{-2}-26 x^{-1}+8=0 \\[4pt]$$ $$8 x^{-2}-2 x^{-1}-3=0 \\[4pt]$$ $$15 x^{-2}-4 x^{-1}-4=0$$ $$4 y^{-2}-9=0 \\[4pt]$$ $$16 y^{-2}+4 y^{-1}=0 \\[4pt]$$ $$2 x^{-2 / 3}-3 x^{-1 / 3}-2=0 \\[4pt]$$ $$4 x^{-1}-17 x^{-1 / 2}+4=0 \\[4pt]$$ $$3 x^{-1}-8 x^{-1 / 2}+4=0$$
 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. $$\left(\dfrac{x-3}{x}\right)^{2}-2\left(\dfrac{x-3}{x}\right)-24=0 \\[4pt]$$ $$\left(\dfrac{2 x+1}{x}\right)^{2}+9\left(\dfrac{2 x+1}{x}\right)-36=0 \\[4pt]$$ $$2\left(\dfrac{x}{x+1}\right)^{2}-5\left(\dfrac{x}{x+1}\right)-3=0 \\[4pt]$$ $$3\left(\dfrac{x}{3 x-1}\right)^{2}+13\left(\dfrac{x}{3 x-1}\right)-10=0 \\[4pt]$$ $$12\left(\dfrac{x}{2 x-3}\right)^{2}-11\left(\dfrac{x}{2 x-3}\right)+2=0 \\[4pt]$$ $$5\left(\dfrac{1}{x+2}\right)^{2}-3\left(\dfrac{1}{x+2}\right)-2=0 \\[4pt]$$ $$\left( \dfrac{2x}{x-5} \right)^{2} +\dfrac{6x}{x-5}-18=0 \\[4pt]$$ $$\dfrac{x^2+2}{x-6}+\dfrac{x-6}{x^2+2} +\dfrac{13}{6}=0$$ $$\dfrac{x^2+2}{x^2-2}+\dfrac{x^2-2}{x^2+2} -\dfrac{170}{77}=0 \\[4pt]$$ $$\sqrt{\dfrac{x+10}{x-6}} + \sqrt{\dfrac{x-6}{x+10} }-\dfrac{34}{15}=0 \\[4pt]$$ $$\sqrt{\dfrac{x+7}{x-5}} - \sqrt{\dfrac{x-5}{x+7} }-\dfrac{3}{2} =0$$
 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. $$\{ 9 \}$$