Mathematics 10 — Exercise 5.4 · Q1 (ii)

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. Let t = x², so x⁴ = t².

   $$t=x^2$$

2. After substitution the quadratic equation is:

   $$4\times t^2-27\times t+18=0$$

3. Substitute the function expressions, keeping the brackets.

   $$4\times t^2-27\times t+18-0$$

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

   $$4\times t^2-27\times t+18$$

5. Write EVERY distributed product before collecting terms.

   $$18-27\times t+4\times t^2-0$$

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

   $$18=18$$

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

   $$-27\times t=-27\times t$$

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

   $$4\times t^2=4\times t^2$$

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

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

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

    $$18+0=18$$

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

    $$-27=-27$$

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

    $$4=4$$

13. Put the simplified groups together.

    $$4\times t^2-27\times t+18$$

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

    $$4\times t^2-27\times t+18$$

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

    $$4\times t^2-27\times t+18=0$$

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

    $$(-3-24=-27,\left(-3\right)\times \left(-24\right)=72)$$

17. Rewrite the middle term using those two parts.

    $$4\times t^2-3\times t-24\times t+18=0$$

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

    $$t\times \left(4\times t-3\right)-6\times \left(4\times t-3\right)=0$$

19. The common bracket can now be factored out.

    $$\left(t-6\right)\times \left(4\times t-3\right)=0$$

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

    $$(t-6=0,4\times t-3=0)$$

21. Take this factor equal to zero.

    $$t-6=0$$

22. Subtract the constant from both sides.

    $$t-6+6=0+6$$

23. Simplify both sides.

    $$t=6$$

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

    $$t=6\times \frac{1}{1}$$

25. Reduce the fraction to obtain this root.

    $$t=6$$

26. Take this factor equal to zero.

    $$4\times t-3=0$$

27. Subtract the constant from both sides.

    $$4\times t-3+3=0+3$$

28. Simplify both sides.

    $$4\times t=3$$

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

    $$t=\frac{3}{4}$$

30. Reduce the fraction to obtain this root.

    $$t=\frac{3}{4}$$

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

    $$0=0$$

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

    $$0=0$$

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 t value back into x²=t.

    $$x^2=\frac{3}{4}$$

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

    $$4\ne 0$$

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

    $$4\times x^2=4\times \frac{3}{4}$$

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

    $$4\times x^2-3=0$$

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

40. Calculate this operation before combining the remaining terms.

    $$16\times \left(-3\right)=-48$$

41. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-48\right)=48$$

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

    $$0^2-16\times \left(-3\right)$$

43. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

44. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 4\times \left(-3\right)=48$$

45. Calculate this operation before combining the remaining terms.

    $$0+48=48$$

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

    $$(a=4,b=0,c=-3)$$

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

    $$D=0^2-16\times \left(-3\right)$$

48. Simplify the discriminant.

    $$D=48$$

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

    $$(x=\frac{0+4\times \sqrt{3}}{8},x=\frac{0-4\times \sqrt{3}}{8})$$

50. Simplify each numerator and divide by 2a.

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

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

    $$x=\frac{0+1\times \sqrt{48}}{2\times 4}$$

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

    $$48=16\times 3$$

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

    $$\sqrt{48}=4\times \sqrt{3}$$

54. Calculate this operation before combining the remaining terms.

    $$\sqrt{48}=4\times \sqrt{3}$$

55. Calculate this operation before combining the remaining terms.

    $$1\times 4\times \sqrt{3}=4\times \sqrt{3}$$

56. Calculate this operation before combining the remaining terms.

    $$0+4\times \sqrt{3}=4\times \sqrt{3}$$

57. Calculate this operation before combining the remaining terms.

    $$2\times 4=8$$

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

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

59. Calculate these two products separately.

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

60. Calculate this operation before combining the remaining terms.

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

61. Calculate this operation before combining the remaining terms.

    $$4\times \sqrt{3}\times \frac{1}{8}=\frac{1}{2}\times \sqrt{3}$$

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

    $$x=\frac{0-\sqrt{48}}{2\times 4}$$

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

    $$48=16\times 3$$

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

    $$\sqrt{48}=4\times \sqrt{3}$$

65. Calculate this operation before combining the remaining terms.

    $$\sqrt{48}=4\times \sqrt{3}$$

66. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4\times \sqrt{3}=\left(-4\right)\times \sqrt{3}$$

67. Calculate this operation before combining the remaining terms.

    $$0-4\times \sqrt{3}=\left(-4\right)\times \sqrt{3}$$

68. Calculate this operation before combining the remaining terms.

    $$2\times 4=8$$

69. Calculate this operation before combining the remaining terms.

    $$\left(\left(-4\right)\times \sqrt{3}\right)\times \frac{1}{8}=\left(-\frac{1}{2}\right)\times \sqrt{3}$$

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

    $$\frac{3}{4}=\frac{3}{4}$$

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

    $$\frac{3}{4}=\frac{3}{4}$$

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

    $$\frac{3}{4}=\frac{3}{4}$$

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

    $$\frac{3}{4}=\frac{3}{4}$$

74. Substitute this t value back into x²=t.

    $$x^2=6$$

75. Substitute the function expressions, keeping the brackets.

    $$x^2-6$$

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

    $$x^2-6$$

77. Write EVERY distributed product before collecting terms.

    $$x^2-6$$

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

    $$x^2=x^2$$

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

    $$\left(-1\right)\times 6=-6$$

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

    $$1=1$$

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

    $$-6=-6$$

82. Put the simplified groups together.

    $$x^2-6$$

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

    $$x^2-6$$

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

    $$x^2-6=0$$

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

86. Calculate this operation before combining the remaining terms.

    $$4\times \left(-6\right)=-24$$

87. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-24\right)=24$$

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

    $$0^2-4\times \left(-6\right)$$

89. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

90. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times \left(-6\right)=24$$

91. Calculate this operation before combining the remaining terms.

    $$0+24=24$$

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

    $$(a=1,b=0,c=-6)$$

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

    $$D=0^2-4\times \left(-6\right)$$

94. Simplify the discriminant.

    $$D=24$$

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

    $$(x=\frac{0+2\times \sqrt{6}}{2},x=\frac{0-2\times \sqrt{6}}{2})$$

96. Simplify each numerator and divide by 2a.

    $$\{\left(-1\right)\times \sqrt{6},\sqrt{6}\}$$

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

    $$x=\frac{0+1\times \sqrt{24}}{2\times 1}$$

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

    $$24=4\times 6$$

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

    $$\sqrt{24}=2\times \sqrt{6}$$

100. Calculate this operation before combining the remaining terms.

     $$\sqrt{24}=2\times \sqrt{6}$$

101. Calculate this operation before combining the remaining terms.

     $$1\times 2\times \sqrt{6}=2\times \sqrt{6}$$

102. Calculate this operation before combining the remaining terms.

     $$0+2\times \sqrt{6}=2\times \sqrt{6}$$

103. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

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

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

105. Calculate these two products separately.

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

106. Calculate this operation before combining the remaining terms.

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

107. Calculate this operation before combining the remaining terms.

     $$2\times \sqrt{6}\times \frac{1}{2}=\sqrt{6}$$

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

     $$x=\frac{0-\sqrt{24}}{2\times 1}$$

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

     $$24=4\times 6$$

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

     $$\sqrt{24}=2\times \sqrt{6}$$

111. Calculate this operation before combining the remaining terms.

     $$\sqrt{24}=2\times \sqrt{6}$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 2\times \sqrt{6}=\left(-2\right)\times \sqrt{6}$$

113. Calculate this operation before combining the remaining terms.

     $$0-2\times \sqrt{6}=\left(-2\right)\times \sqrt{6}$$

114. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

115. Calculate this operation before combining the remaining terms.

     $$\left(\left(-2\right)\times \sqrt{6}\right)\times \frac{1}{2}=\left(-1\right)\times \sqrt{6}$$

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

     $$6=6$$

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

     $$6=6$$

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

     $$6=6$$

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

     $$6=6$$

120. Collect every valid root from every substitution branch.

     $$\{\left(-\frac{1}{2}\right)\times \sqrt{3},\frac{1}{2}\times \sqrt{3},\left(-1\right)\times \sqrt{6},\sqrt{6}\}$$

Answer

The final answer is shown below.