Mathematics 10 — Exercise 5.4 · Q1 (ix)

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

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

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

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

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

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

4. After substitution the quadratic equation is:

   $$2\times t^2-25\times t-13=0$$

5. Substitute the function expressions, keeping the brackets.

   $$-(-2\times t^2+25\times t+13)-0$$

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

   $$2\times t^2-25\times t-13$$

7. Write EVERY distributed product before collecting terms.

   $$-13-25\times t+2\times t^2-0$$

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

   $$-13=-13$$

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

   $$-25\times t=-25\times t$$

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

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

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

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

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

    $$-13+0=-13$$

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

    $$-25=-25$$

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

    $$2=2$$

15. Put the simplified groups together.

    $$2\times t^2-25\times t-13$$

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

    $$2\times t^2-25\times t-13$$

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

    $$2\times t^2-25\times t-13=0$$

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

    $$(1-26=-25,1\times \left(-26\right)=-26)$$

19. Rewrite the middle term using those two parts.

    $$2\times t^2+t-26\times t-13=0$$

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

    $$t\times \left(2\times t+1\right)-13\times \left(2\times t+1\right)=0$$

21. The common bracket can now be factored out.

    $$\left(t-13\right)\times \left(2\times t+1\right)=0$$

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

    $$(t-13=0,2\times t+1=0)$$

23. Take this factor equal to zero.

    $$t-13=0$$

24. Subtract the constant from both sides.

    $$t-13+13=0+13$$

25. Simplify both sides.

    $$t=13$$

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

    $$t=13\times \frac{1}{1}$$

27. Reduce the fraction to obtain this root.

    $$t=13$$

28. Take this factor equal to zero.

    $$2\times t+1=0$$

29. Subtract the constant from both sides.

    $$2\times t+1-1=0-1$$

30. Simplify both sides.

    $$2\times t=-1$$

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

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

32. Reduce the fraction to obtain this root.

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

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

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

    $$0=0$$

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

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

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

    $$2\ne 0$$

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

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

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

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

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

42. Calculate this operation before combining the remaining terms.

    $$8\times \left(-2\right)=-16$$

43. Calculate this operation before combining the remaining terms.

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

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

    $$1^2-8\times \left(-2\right)$$

45. Calculate this operation before combining the remaining terms.

    $$1^2=1$$

46. Calculate this operation before combining the remaining terms.

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

47. Calculate this operation before combining the remaining terms.

    $$1+16=17$$

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

    $$(a=2,b=1,c=-2)$$

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

    $$D=1^2-8\times \left(-2\right)$$

50. Simplify the discriminant.

    $$D=17$$

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

    $$(x=\frac{-1+\sqrt{17}}{4},x=\frac{\left(-1\right)\times \sqrt{17}-1}{4})$$

52. Simplify each numerator and divide by 2a.

    $$\{-\frac{1}{4}+\frac{1}{4}\times \sqrt{17},\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}\}$$

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

    $$x=\frac{-1+1\times \sqrt{17}}{2\times 2}$$

54. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{17}=\sqrt{17}$$

55. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

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

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

57. Calculate these two products separately.

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

58. Calculate this operation before combining the remaining terms.

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

59. Calculate this operation before combining the remaining terms.

    $$-1+\sqrt{17}\times \frac{1}{4}=-\frac{1}{4}+\frac{1}{4}\times \sqrt{17}$$

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

    $$x=\frac{\left(-1\right)\times \sqrt{17}-1}{2\times 2}$$

61. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

62. Calculate this operation before combining the remaining terms.

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

63. Calculate this operation before combining the remaining terms.

    $$-\sqrt{17}-1=\left(-1\right)\times \sqrt{17}-1$$

64. Calculate this operation before combining the remaining terms.

    $$\left(\left(-1\right)\times \sqrt{17}-1\right)\times \frac{1}{4}=\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}$$

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

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

66. Calculate these two products separately.

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

67. Calculate this operation before combining the remaining terms.

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

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

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

69. Calculate these two products separately.

    $$\frac{1}{8}$$

70. Calculate this operation before combining the remaining terms.

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

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

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

72. Calculate these two products separately.

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

73. Calculate this operation before combining the remaining terms.

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

74. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{1}{4}+\frac{1}{4}\times \sqrt{17}\right)}^2=\frac{9}{8}-\frac{1}{8}\times \sqrt{17}$$

75. Calculate this operation before combining the remaining terms.

    $$-\frac{9}{8}+\frac{1}{8}\times \sqrt{17}+\frac{9}{8}-\frac{1}{8}\times \sqrt{17}=0$$

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

    $$-\frac{9}{8}+\frac{1}{8}\times \sqrt{17}+{\left(-\frac{1}{4}+\frac{1}{4}\times \sqrt{17}\right)}^2=0$$

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

    $$0=0$$

78. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{8}\times \sqrt{17}=\left(-\frac{1}{8}\right)\times \sqrt{17}$$

79. Calculate this operation before combining the remaining terms.

    $$-\frac{1}{4}\times \sqrt{17}-\frac{1}{4}=\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}$$

80. Calculate this operation before combining the remaining terms.

    $${\left(\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}\right)}^2=\frac{1}{8}\times \sqrt{17}+\frac{9}{8}$$

81. Calculate this operation before combining the remaining terms.

    $$-\frac{9}{8}-\frac{1}{8}\times \sqrt{17}+\frac{1}{8}\times \sqrt{17}+\frac{9}{8}=0$$

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

    $$-\frac{9}{8}-\frac{1}{8}\times \sqrt{17}+{\left(\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}\right)}^2=0$$

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

    $$0=0$$

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

    $$x^2-1=13\times x$$

85. Substitute the function expressions, keeping the brackets.

    $$-(-x^2+13\times x+1)-0$$

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

    $$x^2-13\times x-1$$

87. Write EVERY distributed product before collecting terms.

    $$-1+x^2-13\times x-0$$

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

    $$-1=-1$$

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

    $$x^2=x^2$$

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

    $$-13\times x=-13\times x$$

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

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

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

    $$-1+0=-1$$

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

    $$1=1$$

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

    $$-13=-13$$

95. Put the simplified groups together.

    $$x^2-13\times x-1$$

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

    $$x^2-13\times x-1$$

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

    $$x^2-13\times x-1=0$$

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

99. Calculate this operation before combining the remaining terms.

    $$4\times \left(-1\right)=-4$$

100. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \left(-4\right)=4$$

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

     $${\left(-13\right)}^2-4\times \left(-1\right)$$

102. Calculate this operation before combining the remaining terms.

     $${\left(-13\right)}^2=169$$

103. Calculate this operation before combining the remaining terms.

     $$\left(-4\right)\times 1\times \left(-1\right)=4$$

104. Calculate this operation before combining the remaining terms.

     $$169+4=173$$

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

     $$(a=1,b=-13,c=-1)$$

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

     $$D={\left(-13\right)}^2-4\times \left(-1\right)$$

107. Simplify the discriminant.

     $$D=173$$

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

     $$(x=\frac{13+\sqrt{173}}{2},x=\frac{13-\sqrt{173}}{2})$$

109. Simplify each numerator and divide by 2a.

     $$\{\frac{13}{2}-\frac{1}{2}\times \sqrt{173},\frac{13}{2}+\frac{1}{2}\times \sqrt{173}\}$$

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

     $$x=\frac{13+1\times \sqrt{173}}{2\times 1}$$

111. Calculate this operation before combining the remaining terms.

     $$1\times \sqrt{173}=\sqrt{173}$$

112. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

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

     $$\frac{1\times 1}{1\times 2}$$

114. Calculate these two products separately.

     $$\frac{1}{2}$$

115. Calculate this operation before combining the remaining terms.

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

116. Calculate this operation before combining the remaining terms.

     $$13+\sqrt{173}\times \frac{1}{2}=\frac{13}{2}+\frac{1}{2}\times \sqrt{173}$$

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

     $$x=\frac{13-\sqrt{173}}{2\times 1}$$

118. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

119. Calculate this operation before combining the remaining terms.

     $$\left(13-\sqrt{173}\right)\times \frac{1}{2}=\frac{13}{2}-\frac{1}{2}\times \sqrt{173}$$

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

     $$\frac{\left(-171\right)\times 1}{1\times 2}$$

121. Calculate these two products separately.

     $$-\frac{171}{2}$$

122. Calculate this operation before combining the remaining terms.

     $$\left(-171\right)\times \frac{1}{2}=-\frac{171}{2}$$

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

     $$\frac{13\times 1}{1\times 2}$$

124. Calculate these two products separately.

     $$\frac{13}{2}$$

125. Calculate this operation before combining the remaining terms.

     $$13\times \frac{1}{2}=\frac{13}{2}$$

126. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{13}{2}\times \sqrt{173}=\left(-\frac{13}{2}\right)\times \sqrt{173}$$

127. Calculate this operation before combining the remaining terms.

     $${\left(\frac{13}{2}+\frac{1}{2}\times \sqrt{173}\right)}^2=\frac{1}{4}\times {\left(13+\sqrt{173}\right)}^2$$

128. Calculate this operation before combining the remaining terms.

     $$-\frac{171}{2}-\frac{13}{2}\times \sqrt{173}+\frac{1}{4}\times {\left(13+\sqrt{173}\right)}^2=0$$

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

     $$-\frac{171}{2}-\frac{13}{2}\times \sqrt{173}+{\left(\frac{13}{2}+\frac{1}{2}\times \sqrt{173}\right)}^2=0$$

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

     $$0=0$$

131. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{2}\times \sqrt{173}=\left(-\frac{1}{2}\right)\times \sqrt{173}$$

132. Calculate this operation before combining the remaining terms.

     $${\left(\frac{13}{2}-\frac{1}{2}\times \sqrt{173}\right)}^2=\frac{1}{4}\times {\left(13-\sqrt{173}\right)}^2$$

133. Calculate this operation before combining the remaining terms.

     $$-\frac{171}{2}+\frac{1}{4}\times {\left(13-\sqrt{173}\right)}^2+\frac{13}{2}\times \sqrt{173}=0$$

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

     $$-\frac{171}{2}+{\left(\frac{13}{2}-\frac{1}{2}\times \sqrt{173}\right)}^2+\frac{13}{2}\times \sqrt{173}=0$$

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

     $$0=0$$

136. Collect every valid root from every substitution branch.

     $$\{-\frac{1}{4}+\frac{1}{4}\times \sqrt{17},\frac{13}{2}-\frac{1}{2}\times \sqrt{173},\frac{13}{2}+\frac{1}{2}\times \sqrt{173},\left(-\frac{1}{4}\right)\times \sqrt{17}-\frac{1}{4}\}$$

Answer

The final answer is shown below.