Mathematics 10 — Exercise 5.4 · Q1 (x)

Full question

Solve the following equation by reducing it to a quadratic equation. Check the resulting values in the original equation.

Step-by-step solution

1. x=0 is not a solution. Divide by x² and group: 2(x²+1/x²)−5(x+1/x)−14=0.

   $$2\times x^2-5\times x-14-\frac{5}{x}+\frac{2}{x^2}=0$$

2. The reciprocal requires x ≠ 0. Let t equal the indicated reciprocal sum/difference.

   $$t=x+\frac{1}{x}$$

3. Square that definition: the cross term is +2 for a sum and −2 for a difference.

   $$t^2=x^2+2+\frac{1}{x^2}$$

4. Rearrange to replace x²+1/x² by t² minus the cross term.

   $$x^2+\frac{1}{x^2}=t^2-2$$

5. After substitution the quadratic equation is:

   $$2\times t^2-5\times t-18=0$$

6. Substitute the function expressions, keeping the brackets.

   $$-(-2\times t^2+5\times t+18)-0$$

7. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$2\times t^2-5\times t-18$$

8. Write EVERY distributed product before collecting terms.

   $$-18-5\times t+2\times t^2-0$$

9. Calculate this individual product, including its sign and powers.

   $$-18=-18$$

10. Calculate this individual product, including its sign and powers.

    $$-5\times t=-5\times t$$

11. Calculate this individual product, including its sign and powers.

    $$2\times t^2=2\times t^2$$

12. Calculate this individual product, including its sign and powers.

    $$\left(-1\right)\times 0=0$$

13. These terms have the same variable part. Add ONLY their coefficients.

    $$-18+0=-18$$

14. These terms have the same variable part. Add ONLY their coefficients.

    $$-5=-5$$

15. These terms have the same variable part. Add ONLY their coefficients.

    $$2=2$$

16. Put the simplified groups together.

    $$2\times t^2-5\times t-18$$

17. Collect terms with the same power and simplify their coefficients.

    $$2\times t^2-5\times t-18$$

18. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times t^2-5\times t-18=0$$

19. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(-9+4=-5,\left(-9\right)\times 4=-36)$$

20. Rewrite the middle term using those two parts.

    $$2\times t^2-9\times t+4\times t-18=0$$

21. Group the first two terms and the last two. Take out the common factor from each group.

    $$t\times \left(2\times t-9\right)+2\times \left(2\times t-9\right)=0$$

22. The common bracket can now be factored out.

    $$\left(t+2\right)\times \left(2\times t-9\right)=0$$

23. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(t+2=0,2\times t-9=0)$$

24. Take this factor equal to zero.

    $$t+2=0$$

25. Subtract the constant from both sides.

    $$t+2-2=0-2$$

26. Simplify both sides.

    $$t=-2$$

27. Divide both sides by the coefficient of the variable.

    $$t=-2\times \frac{1}{1}$$

28. Reduce the fraction to obtain this root.

    $$t=-2$$

29. Take this factor equal to zero.

    $$2\times t-9=0$$

30. Subtract the constant from both sides.

    $$2\times t-9+9=0+9$$

31. Simplify both sides.

    $$2\times t=9$$

32. Divide both sides by the coefficient of the variable.

    $$t=\frac{9}{2}$$

33. Reduce the fraction to obtain this root.

    $$t=\frac{9}{2}$$

34. Substitute this root into the original equation before simplifying.

    $$0=0$$

35. Calculate both sides; their equal values confirm the root.

    $$0=0$$

36. Substitute this root into the original equation before simplifying.

    $$0=0$$

37. Calculate both sides; their equal values confirm the root.

    $$0=0$$

38. Restore the reciprocal definition and multiply both sides by nonzero x.

    $$x^2+1=-2\times x$$

39. Substitute the function expressions, keeping the brackets.

    $$x^2+2\times x+1-0$$

40. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

    $$x^2+2\times x+1$$

41. Write EVERY distributed product before collecting terms.

    $$1+x^2+2\times x-0$$

42. Calculate this individual product, including its sign and powers.

    $$1=1$$

43. Calculate this individual product, including its sign and powers.

    $$x^2=x^2$$

44. Calculate this individual product, including its sign and powers.

    $$2\times x=2\times x$$

45. Calculate this individual product, including its sign and powers.

    $$\left(-1\right)\times 0=0$$

46. These terms have the same variable part. Add ONLY their coefficients.

    $$1+0=1$$

47. These terms have the same variable part. Add ONLY their coefficients.

    $$1=1$$

48. These terms have the same variable part. Add ONLY their coefficients.

    $$2=2$$

49. Put the simplified groups together.

    $$x^2+2\times x+1$$

50. Collect terms with the same power and simplify their coefficients.

    $$x^2+2\times x+1$$

51. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$x^2+2\times x+1=0$$

52. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(1+1=2,1\times 1=1)$$

53. Rewrite the middle term using those two parts.

    $$x^2+x+x+1=0$$

54. Group the first two terms and the last two. Take out the common factor from each group.

    $$x\times \left(x+1\right)+x+1=0$$

55. The common bracket can now be factored out.

    $$\left(x+1\right)\left(x+1\right)=0$$

56. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(x+1=0,x+1=0)$$

57. Take this factor equal to zero.

    $$x+1=0$$

58. Subtract the constant from both sides.

    $$x+1-1=0-1$$

59. Simplify both sides.

    $$x=-1$$

60. Divide both sides by the coefficient of the variable.

    $$x=-\frac{1}{1}$$

61. Reduce the fraction to obtain this root.

    $$x=-1$$

62. Take this factor equal to zero.

    $$x+1=0$$

63. Subtract the constant from both sides.

    $$x+1-1=0-1$$

64. Simplify both sides.

    $$x=-1$$

65. Divide both sides by the coefficient of the variable.

    $$x=-\frac{1}{1}$$

66. Reduce the fraction to obtain this root.

    $$x=-1$$

67. Substitute this root into the original equation before simplifying.

    $$0=0$$

68. Calculate both sides; their equal values confirm the root.

    $$0=0$$

69. Restore the reciprocal definition and multiply both sides by nonzero x.

    $$x^2+1=\frac{9\times x}{2}$$

70. A denominator cannot equal zero. Keep the original excluded values throughout the calculation.

    $$2\ne 0$$

71. Multiply BOTH sides by the common denominator to remove the fractions.

    $$2\times \left(x^2-\frac{9\times x}{2}+1\right)=2\times 0$$

72. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times x^2-9\times x+2=0$$

73. Integer factorization does not apply here, so use the quadratic formula.

74. Calculate this operation before combining the remaining terms.

    $$8\times 2=16$$

75. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 16=-16$$

76. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

    $${\left(-9\right)}^2-8\times 2$$

77. Calculate this operation before combining the remaining terms.

    $${\left(-9\right)}^2=81$$

78. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 2\times 2=-16$$

79. Calculate this operation before combining the remaining terms.

    $$81-16=65$$

80. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

    $$(a=2,b=-9,c=2)$$

81. Compute D = b² − 4ac. A minus times a minus becomes a plus.

    $$D={\left(-9\right)}^2-8\times 2$$

82. Simplify the discriminant.

    $$D=65$$

83. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

    $$(x=\frac{\sqrt{65}+9}{4},x=\frac{9-\sqrt{65}}{4})$$

84. Simplify each numerator and divide by 2a.

    $$\{\frac{9}{4}-\frac{1}{4}\times \sqrt{65},\frac{1}{4}\times \sqrt{65}+\frac{9}{4}\}$$

85. Calculate this root branch separately, preserving the sign.

    $$x=\frac{1\times \sqrt{65}+9}{2\times 2}$$

86. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{65}=\sqrt{65}$$

87. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

88. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 4}$$

89. Calculate these two products separately.

    $$\frac{1}{4}$$

90. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{4}=\frac{1}{4}$$

91. Calculate this operation before combining the remaining terms.

    $$\sqrt{65}+9\times \frac{1}{4}=\frac{1}{4}\times \sqrt{65}+\frac{9}{4}$$

92. Calculate this root branch separately, preserving the sign.

    $$x=\frac{9-\sqrt{65}}{2\times 2}$$

93. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

94. Calculate this operation before combining the remaining terms.

    $$9-\sqrt{65}\times \frac{1}{4}=\frac{9}{4}-\frac{1}{4}\times \sqrt{65}$$

95. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-73\right)\times 1}{1\times 8}$$

96. Calculate these two products separately.

    $$-\frac{73}{8}$$

97. Calculate this operation before combining the remaining terms.

    $$\left(-73\right)\times \frac{1}{8}=-\frac{73}{8}$$

98. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{9\times 1}{1\times 8}$$

99. Calculate these two products separately.

    $$\frac{9}{8}$$

100. Calculate this operation before combining the remaining terms.

     $$9\times \frac{1}{8}=\frac{9}{8}$$

101. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{9}{8}\times \sqrt{65}=\left(-\frac{9}{8}\right)\times \sqrt{65}$$

102. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{9\times 1}{1\times 4}$$

103. Calculate these two products separately.

     $$\frac{9}{4}$$

104. Calculate this operation before combining the remaining terms.

     $$9\times \frac{1}{4}=\frac{9}{4}$$

105. Calculate this operation before combining the remaining terms.

     $${\left(\frac{1}{4}\times \sqrt{65}+\frac{9}{4}\right)}^2=\frac{1}{16}\times {\left(\sqrt{65}+9\right)}^2$$

106. Calculate this operation before combining the remaining terms.

     $$-\frac{73}{8}-\frac{9}{8}\times \sqrt{65}+\frac{1}{16}\times {\left(\sqrt{65}+9\right)}^2=0$$

107. Substitute this root into the original equation before simplifying.

     $$-\frac{73}{8}-\frac{9}{8}\times \sqrt{65}+{\left(\frac{1}{4}\times \sqrt{65}+\frac{9}{4}\right)}^2=0$$

108. Calculate both sides; their equal values confirm the root.

     $$0=0$$

109. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{4}\times \sqrt{65}=\left(-\frac{1}{4}\right)\times \sqrt{65}$$

110. Calculate this operation before combining the remaining terms.

     $${\left(\frac{9}{4}-\frac{1}{4}\times \sqrt{65}\right)}^2=\frac{1}{16}\times {\left(9-\sqrt{65}\right)}^2$$

111. Calculate this operation before combining the remaining terms.

     $$-\frac{73}{8}+\frac{1}{16}\times {\left(9-\sqrt{65}\right)}^2+\frac{9}{8}\times \sqrt{65}=0$$

112. Substitute this root into the original equation before simplifying.

     $$-\frac{73}{8}+{\left(\frac{9}{4}-\frac{1}{4}\times \sqrt{65}\right)}^2+\frac{9}{8}\times \sqrt{65}=0$$

113. Calculate both sides; their equal values confirm the root.

     $$0=0$$

114. Collect every valid root from every substitution branch.

     $$\{-1,\frac{9}{4}-\frac{1}{4}\times \sqrt{65},\frac{1}{4}\times \sqrt{65}+\frac{9}{4}\}$$

Answer

The final answer is shown below.