Mathematics 10 — Exercise Review · Q4 (i)

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:

   $$t^2-16\times t+63=0$$

3. Substitute the function expressions, keeping the brackets.

   $$t^2-16\times t+63-0$$

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

   $$t^2-16\times t+63$$

5. Write EVERY distributed product before collecting terms.

   $$63+t^2-16\times t-0$$

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

   $$63=63$$

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

   $$t^2=t^2$$

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

   $$-16\times t=-16\times t$$

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.

    $$63+0=63$$

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

    $$1=1$$

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

    $$-16=-16$$

13. Put the simplified groups together.

    $$t^2-16\times t+63$$

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

    $$t^2-16\times t+63$$

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

    $$t^2-16\times t+63=0$$

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

    $$(-7-9=-16,\left(-7\right)\times \left(-9\right)=63)$$

17. Rewrite the middle term using those two parts.

    $$t^2-7\times t-9\times t+63=0$$

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

    $$t\times \left(t-7\right)-9\times \left(t-7\right)=0$$

19. The common bracket can now be factored out.

    $$\left(t-9\right)\times \left(t-7\right)=0$$

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

    $$(t-9=0,t-7=0)$$

21. Take this factor equal to zero.

    $$t-9=0$$

22. Subtract the constant from both sides.

    $$t-9+9=0+9$$

23. Simplify both sides.

    $$t=9$$

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

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

25. Reduce the fraction to obtain this root.

    $$t=9$$

26. Take this factor equal to zero.

    $$t-7=0$$

27. Subtract the constant from both sides.

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

28. Simplify both sides.

    $$t=7$$

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

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

30. Reduce the fraction to obtain this root.

    $$t=7$$

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.

    $$x^2=7$$

36. Substitute the function expressions, keeping the brackets.

    $$x^2-7$$

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

    $$x^2-7$$

38. Write EVERY distributed product before collecting terms.

    $$x^2-7$$

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

    $$x^2=x^2$$

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

    $$\left(-1\right)\times 7=-7$$

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

    $$1=1$$

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

    $$-7=-7$$

43. Put the simplified groups together.

    $$x^2-7$$

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

    $$x^2-7$$

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

    $$x^2-7=0$$

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

47. Calculate this operation before combining the remaining terms.

    $$4\times \left(-7\right)=-28$$

48. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-28\right)=28$$

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

    $$0^2-4\times \left(-7\right)$$

50. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

51. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times \left(-7\right)=28$$

52. Calculate this operation before combining the remaining terms.

    $$0+28=28$$

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

    $$(a=1,b=0,c=-7)$$

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

    $$D=0^2-4\times \left(-7\right)$$

55. Simplify the discriminant.

    $$D=28$$

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

    $$(x=\frac{0+2\times \sqrt{7}}{2},x=\frac{0-2\times \sqrt{7}}{2})$$

57. Simplify each numerator and divide by 2a.

    $$\{\left(-1\right)\times \sqrt{7},\sqrt{7}\}$$

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

    $$x=\frac{0+1\times \sqrt{28}}{2\times 1}$$

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

    $$28=4\times 7$$

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

    $$\sqrt{28}=2\times \sqrt{7}$$

61. Calculate this operation before combining the remaining terms.

    $$\sqrt{28}=2\times \sqrt{7}$$

62. Calculate this operation before combining the remaining terms.

    $$1\times 2\times \sqrt{7}=2\times \sqrt{7}$$

63. Calculate this operation before combining the remaining terms.

    $$0+2\times \sqrt{7}=2\times \sqrt{7}$$

64. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

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

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

66. Calculate these two products separately.

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

67. Calculate this operation before combining the remaining terms.

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

68. Calculate this operation before combining the remaining terms.

    $$2\times \sqrt{7}\times \frac{1}{2}=\sqrt{7}$$

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

    $$x=\frac{0-\sqrt{28}}{2\times 1}$$

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

    $$28=4\times 7$$

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

    $$\sqrt{28}=2\times \sqrt{7}$$

72. Calculate this operation before combining the remaining terms.

    $$\sqrt{28}=2\times \sqrt{7}$$

73. Calculate this operation before combining the remaining terms.

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

74. Calculate this operation before combining the remaining terms.

    $$0-2\times \sqrt{7}=\left(-2\right)\times \sqrt{7}$$

75. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

76. Calculate this operation before combining the remaining terms.

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

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

    $$7=7$$

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

    $$7=7$$

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

    $$7=7$$

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

    $$7=7$$

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

    $$x^2=9$$

82. Substitute the function expressions, keeping the brackets.

    $$x^2-9$$

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

    $$x^2-9$$

84. Write EVERY distributed product before collecting terms.

    $$x^2-9$$

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

    $$x^2=x^2$$

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

    $$\left(-1\right)\times 9=-9$$

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

    $$1=1$$

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

    $$-9=-9$$

89. Put the simplified groups together.

    $$x^2-9$$

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

    $$x^2-9$$

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

    $$x^2-9=0$$

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

    $$(3-3=0,3\times \left(-3\right)=-9)$$

93. Rewrite the middle term using those two parts.

    $$x^2+3\times x-3\times x-9=0$$

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

    $$x\times \left(x+3\right)-3\times \left(x+3\right)=0$$

95. The common bracket can now be factored out.

    $$\left(x-3\right)\times \left(x+3\right)=0$$

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

    $$(x-3=0,x+3=0)$$

97. Take this factor equal to zero.

    $$x-3=0$$

98. Subtract the constant from both sides.

    $$x-3+3=0+3$$

99. Simplify both sides.

    $$x=3$$

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

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

101. Reduce the fraction to obtain this root.

     $$x=3$$

102. Take this factor equal to zero.

     $$x+3=0$$

103. Subtract the constant from both sides.

     $$x+3-3=0-3$$

104. Simplify both sides.

     $$x=-3$$

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

     $$x=-3\times \frac{1}{1}$$

106. Reduce the fraction to obtain this root.

     $$x=-3$$

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

     $$9=9$$

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

     $$9=9$$

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

     $$9=9$$

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

     $$9=9$$

111. Collect every valid root from every substitution branch.

     $$\{-3,3,\left(-1\right)\times \sqrt{7},\sqrt{7}\}$$

Answer

The final answer is shown below.