Mathematics 10 — Exercise 5.4 · Q1 (viii)

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:

   $$t^2+2\times t-35=0$$

5. Substitute the function expressions, keeping the brackets.

   $$t^2+2\times t-35-0$$

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

   $$t^2+2\times t-35$$

7. Write EVERY distributed product before collecting terms.

   $$-35+t^2+2\times t-0$$

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

   $$-35=-35$$

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

   $$t^2=t^2$$

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

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

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.

    $$-35+0=-35$$

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

    $$1=1$$

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

    $$2=2$$

15. Put the simplified groups together.

    $$t^2+2\times t-35$$

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

    $$t^2+2\times t-35$$

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

    $$t^2+2\times t-35=0$$

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

    $$(7-5=2,7\times \left(-5\right)=-35)$$

19. Rewrite the middle term using those two parts.

    $$t^2+7\times t-5\times t-35=0$$

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

    $$t\times \left(t+7\right)-5\times \left(t+7\right)=0$$

21. The common bracket can now be factored out.

    $$\left(t-5\right)\times \left(t+7\right)=0$$

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

    $$(t-5=0,t+7=0)$$

23. Take this factor equal to zero.

    $$t-5=0$$

24. Subtract the constant from both sides.

    $$t-5+5=0+5$$

25. Simplify both sides.

    $$t=5$$

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

    $$t=5\times \frac{1}{1}$$

27. Reduce the fraction to obtain this root.

    $$t=5$$

28. Take this factor equal to zero.

    $$t+7=0$$

29. Subtract the constant from both sides.

    $$t+7-7=0-7$$

30. Simplify both sides.

    $$t=-7$$

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

    $$t=-7\times \frac{1}{1}$$

32. Reduce the fraction to obtain this root.

    $$t=-7$$

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.

    $$x^2+1=-7\times x$$

38. Substitute the function expressions, keeping the brackets.

    $$x^2+7\times x+1-0$$

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

    $$x^2+7\times x+1$$

40. Write EVERY distributed product before collecting terms.

    $$1+x^2+7\times x-0$$

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

    $$1=1$$

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

    $$x^2=x^2$$

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

    $$7\times x=7\times x$$

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

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

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

    $$1+0=1$$

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

    $$1=1$$

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

    $$7=7$$

48. Put the simplified groups together.

    $$x^2+7\times x+1$$

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

    $$x^2+7\times x+1$$

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

    $$x^2+7\times x+1=0$$

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

52. Calculate this operation before combining the remaining terms.

    $$4\times 1=4$$

53. Calculate this operation before combining the remaining terms.

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

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

    $$7^2-4\times 1$$

55. Calculate this operation before combining the remaining terms.

    $$7^2=49$$

56. Calculate this operation before combining the remaining terms.

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

57. Calculate this operation before combining the remaining terms.

    $$49-4=45$$

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

    $$(a=1,b=7,c=1)$$

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

    $$D=7^2-4\times 1$$

60. Simplify the discriminant.

    $$D=45$$

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

    $$(x=\frac{-7+3\times \sqrt{5}}{2},x=\frac{-7-3\times \sqrt{5}}{2})$$

62. Simplify each numerator and divide by 2a.

    $$\{-\frac{7}{2}-\frac{3}{2}\times \sqrt{5},-\frac{7}{2}+\frac{3}{2}\times \sqrt{5}\}$$

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

    $$x=\frac{-7+1\times \sqrt{45}}{2\times 1}$$

64. Extract a perfect-square factor from under the root.

    $$45=9\times 5$$

65. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{45}=3\times \sqrt{5}$$

66. Calculate this operation before combining the remaining terms.

    $$\sqrt{45}=3\times \sqrt{5}$$

67. Calculate this operation before combining the remaining terms.

    $$1\times 3\times \sqrt{5}=3\times \sqrt{5}$$

68. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

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

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

70. Calculate these two products separately.

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

71. Calculate this operation before combining the remaining terms.

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

72. Calculate this operation before combining the remaining terms.

    $$\left(-7+3\times \sqrt{5}\right)\times \frac{1}{2}=-\frac{7}{2}+\frac{3}{2}\times \sqrt{5}$$

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

    $$x=\frac{-7-\sqrt{45}}{2\times 1}$$

74. Extract a perfect-square factor from under the root.

    $$45=9\times 5$$

75. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{45}=3\times \sqrt{5}$$

76. Calculate this operation before combining the remaining terms.

    $$\sqrt{45}=3\times \sqrt{5}$$

77. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 3\times \sqrt{5}=\left(-3\right)\times \sqrt{5}$$

78. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

79. Calculate this operation before combining the remaining terms.

    $$\left(-7-3\times \sqrt{5}\right)\times \frac{1}{2}=-\frac{7}{2}-\frac{3}{2}\times \sqrt{5}$$

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

    $$\frac{\left(-47\right)\times 1}{1\times 2}$$

81. Calculate these two products separately.

    $$-\frac{47}{2}$$

82. Calculate this operation before combining the remaining terms.

    $$\left(-47\right)\times \frac{1}{2}=-\frac{47}{2}$$

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

    $$\frac{\left(-7\right)\times 1}{1\times 2}$$

84. Calculate these two products separately.

    $$-\frac{7}{2}$$

85. Calculate this operation before combining the remaining terms.

    $$\left(-7\right)\times \frac{1}{2}=-\frac{7}{2}$$

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

    $$\frac{3\times 1}{1\times 2}$$

87. Calculate these two products separately.

    $$\frac{3}{2}$$

88. Calculate this operation before combining the remaining terms.

    $$3\times \frac{1}{2}=\frac{3}{2}$$

89. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{7}{2}+\frac{3}{2}\times \sqrt{5}\right)}^2=\frac{47}{2}-\frac{21}{2}\times \sqrt{5}$$

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

    $$\frac{21\times 1}{1\times 2}$$

91. Calculate these two products separately.

    $$\frac{21}{2}$$

92. Calculate this operation before combining the remaining terms.

    $$21\times \frac{1}{2}=\frac{21}{2}$$

93. Calculate this operation before combining the remaining terms.

    $$-\frac{47}{2}+\frac{47}{2}-\frac{21}{2}\times \sqrt{5}+\frac{21}{2}\times \sqrt{5}=0$$

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

    $$-\frac{47}{2}+{\left(-\frac{7}{2}+\frac{3}{2}\times \sqrt{5}\right)}^2+\frac{21}{2}\times \sqrt{5}=0$$

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

    $$0=0$$

96. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{21}{2}\times \sqrt{5}=\left(-\frac{21}{2}\right)\times \sqrt{5}$$

97. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{3}{2}\times \sqrt{5}=\left(-\frac{3}{2}\right)\times \sqrt{5}$$

98. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{7}{2}-\frac{3}{2}\times \sqrt{5}\right)}^2=\frac{21}{2}\times \sqrt{5}+\frac{47}{2}$$

99. Calculate this operation before combining the remaining terms.

    $$-\frac{47}{2}-\frac{21}{2}\times \sqrt{5}+\frac{21}{2}\times \sqrt{5}+\frac{47}{2}=0$$

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

     $$-\frac{47}{2}-\frac{21}{2}\times \sqrt{5}+{\left(-\frac{7}{2}-\frac{3}{2}\times \sqrt{5}\right)}^2=0$$

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

     $$0=0$$

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

     $$x^2+1=5\times x$$

103. Substitute the function expressions, keeping the brackets.

     $$x^2-5\times x+1-0$$

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

     $$x^2-5\times x+1$$

105. Write EVERY distributed product before collecting terms.

     $$1+x^2-5\times x-0$$

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

     $$1=1$$

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

     $$x^2=x^2$$

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

     $$-5\times x=-5\times x$$

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

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

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

     $$1+0=1$$

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

     $$1=1$$

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

     $$-5=-5$$

113. Put the simplified groups together.

     $$x^2-5\times x+1$$

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

     $$x^2-5\times x+1$$

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

     $$x^2-5\times x+1=0$$

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

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

     $${\left(-5\right)}^2-4\times 1$$

118. Calculate this operation before combining the remaining terms.

     $${\left(-5\right)}^2=25$$

119. Calculate this operation before combining the remaining terms.

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

120. Calculate this operation before combining the remaining terms.

     $$25-4=21$$

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

     $$(a=1,b=-5,c=1)$$

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

     $$D={\left(-5\right)}^2-4\times 1$$

123. Simplify the discriminant.

     $$D=21$$

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

     $$(x=\frac{\sqrt{21}+5}{2},x=\frac{5-\sqrt{21}}{2})$$

125. Simplify each numerator and divide by 2a.

     $$\{\frac{5}{2}-\frac{1}{2}\times \sqrt{21},\frac{1}{2}\times \sqrt{21}+\frac{5}{2}\}$$

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

     $$x=\frac{1\times \sqrt{21}+5}{2\times 1}$$

127. Calculate this operation before combining the remaining terms.

     $$1\times \sqrt{21}=\sqrt{21}$$

128. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

129. Calculate this operation before combining the remaining terms.

     $$\sqrt{21}+5\times \frac{1}{2}=\frac{1}{2}\times \sqrt{21}+\frac{5}{2}$$

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

     $$x=\frac{5-\sqrt{21}}{2\times 1}$$

131. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

132. Calculate this operation before combining the remaining terms.

     $$5-\sqrt{21}\times \frac{1}{2}=\frac{5}{2}-\frac{1}{2}\times \sqrt{21}$$

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

     $$\frac{\left(-23\right)\times 1}{1\times 2}$$

134. Calculate these two products separately.

     $$-\frac{23}{2}$$

135. Calculate this operation before combining the remaining terms.

     $$\left(-23\right)\times \frac{1}{2}=-\frac{23}{2}$$

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

     $$\frac{5\times 1}{1\times 2}$$

137. Calculate these two products separately.

     $$\frac{5}{2}$$

138. Calculate this operation before combining the remaining terms.

     $$5\times \frac{1}{2}=\frac{5}{2}$$

139. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{5}{2}\times \sqrt{21}=\left(-\frac{5}{2}\right)\times \sqrt{21}$$

140. Calculate this operation before combining the remaining terms.

     $${\left(\frac{1}{2}\times \sqrt{21}+\frac{5}{2}\right)}^2=\frac{1}{4}\times {\left(\sqrt{21}+5\right)}^2$$

141. Calculate this operation before combining the remaining terms.

     $$-\frac{23}{2}-\frac{5}{2}\times \sqrt{21}+\frac{1}{4}\times {\left(\sqrt{21}+5\right)}^2=0$$

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

     $$-\frac{23}{2}-\frac{5}{2}\times \sqrt{21}+{\left(\frac{1}{2}\times \sqrt{21}+\frac{5}{2}\right)}^2=0$$

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

     $$0=0$$

144. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{2}\times \sqrt{21}=\left(-\frac{1}{2}\right)\times \sqrt{21}$$

145. Calculate this operation before combining the remaining terms.

     $${\left(\frac{5}{2}-\frac{1}{2}\times \sqrt{21}\right)}^2=\frac{1}{4}\times {\left(5-\sqrt{21}\right)}^2$$

146. Calculate this operation before combining the remaining terms.

     $$-\frac{23}{2}+\frac{1}{4}\times {\left(5-\sqrt{21}\right)}^2+\frac{5}{2}\times \sqrt{21}=0$$

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

     $$-\frac{23}{2}+{\left(\frac{5}{2}-\frac{1}{2}\times \sqrt{21}\right)}^2+\frac{5}{2}\times \sqrt{21}=0$$

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

     $$0=0$$

149. Collect every valid root from every substitution branch.

     $$\{-\frac{7}{2}-\frac{3}{2}\times \sqrt{5},-\frac{7}{2}+\frac{3}{2}\times \sqrt{5},\frac{5}{2}-\frac{1}{2}\times \sqrt{21},\frac{1}{2}\times \sqrt{21}+\frac{5}{2}\}$$

Answer

The final answer is shown below.