Mathematics 10 — Exercise Review · Q4 (vii)

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. Pair the four factors so that each pair has the same sum of offsets.

   $$\left(\left(x-8\right)\times \left(x+5\right)\right)\times \left(\left(x-2\right)\times \left(x-1\right)\right)$$

2. Expand the first pair.

   $$\left(x-2\right)\times \left(x-1\right)=x^2-3\times x+2$$

3. Expand the second pair.

   $$\left(x-8\right)\times \left(x+5\right)=x^2-3\times x-40$$

4. Both contain the same x²+sx. Let this common expression be t.

   $$t=x^2-3\times x$$

5. The product is now a quadratic in t.

   $$\left(t-40\right)\times \left(t+2\right)=-360$$

6. After substitution the quadratic equation is:

   $$t^2-38\times t+280=0$$

7. Substitute the function expressions, keeping the brackets.

   $$t^2-38\times t+280-0$$

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

   $$t^2-38\times t+280$$

9. Write EVERY distributed product before collecting terms.

   $$280+t^2-38\times t-0$$

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

    $$280=280$$

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

    $$t^2=t^2$$

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

    $$-38\times t=-38\times t$$

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

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

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

    $$280+0=280$$

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

    $$1=1$$

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

    $$-38=-38$$

17. Put the simplified groups together.

    $$t^2-38\times t+280$$

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

    $$t^2-38\times t+280$$

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

    $$t^2-38\times t+280=0$$

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

    $$(-10-28=-38,\left(-10\right)\times \left(-28\right)=280)$$

21. Rewrite the middle term using those two parts.

    $$t^2-10\times t-28\times t+280=0$$

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

    $$t\times \left(t-10\right)-28\times \left(t-10\right)=0$$

23. The common bracket can now be factored out.

    $$\left(t-28\right)\times \left(t-10\right)=0$$

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

    $$(t-28=0,t-10=0)$$

25. Take this factor equal to zero.

    $$t-28=0$$

26. Subtract the constant from both sides.

    $$t-28+28=0+28$$

27. Simplify both sides.

    $$t=28$$

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

    $$t=28\times \frac{1}{1}$$

29. Reduce the fraction to obtain this root.

    $$t=28$$

30. Take this factor equal to zero.

    $$t-10=0$$

31. Subtract the constant from both sides.

    $$t-10+10=0+10$$

32. Simplify both sides.

    $$t=10$$

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

    $$t=10\times \frac{1}{1}$$

34. Reduce the fraction to obtain this root.

    $$t=10$$

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. Substitute this root into the original equation before simplifying.

    $$0=0$$

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

    $$0=0$$

39. Restore x²+sx=t for this root.

    $$x^2-3\times x=10$$

40. Substitute the function expressions, keeping the brackets.

    $$x^2-3\times x-10$$

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

    $$x^2-3\times x-10$$

42. Write EVERY distributed product before collecting terms.

    $$x^2-3\times x-10$$

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

    $$x^2=x^2$$

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

    $$-3\times x=-3\times x$$

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

    $$\left(-1\right)\times 10=-10$$

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.

    $$-3=-3$$

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

    $$-10=-10$$

49. Put the simplified groups together.

    $$x^2-3\times x-10$$

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

    $$x^2-3\times x-10$$

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

    $$x^2-3\times x-10=0$$

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

    $$(2-5=-3,2\times \left(-5\right)=-10)$$

53. Rewrite the middle term using those two parts.

    $$x^2+2\times x-5\times x-10=0$$

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

    $$x\times \left(x+2\right)-5\times \left(x+2\right)=0$$

55. The common bracket can now be factored out.

    $$\left(x-5\right)\times \left(x+2\right)=0$$

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

    $$(x-5=0,x+2=0)$$

57. Take this factor equal to zero.

    $$x-5=0$$

58. Subtract the constant from both sides.

    $$x-5+5=0+5$$

59. Simplify both sides.

    $$x=5$$

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

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

61. Reduce the fraction to obtain this root.

    $$x=5$$

62. Take this factor equal to zero.

    $$x+2=0$$

63. Subtract the constant from both sides.

    $$x+2-2=0-2$$

64. Simplify both sides.

    $$x=-2$$

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

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

66. Reduce the fraction to obtain this root.

    $$x=-2$$

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

    $$10=10$$

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

    $$10=10$$

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

    $$10=10$$

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

    $$10=10$$

71. Restore x²+sx=t for this root.

    $$x^2-3\times x=28$$

72. Substitute the function expressions, keeping the brackets.

    $$x^2-3\times x-28$$

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

    $$x^2-3\times x-28$$

74. Write EVERY distributed product before collecting terms.

    $$x^2-3\times x-28$$

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

    $$x^2=x^2$$

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

    $$-3\times x=-3\times x$$

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

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

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

    $$1=1$$

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

    $$-3=-3$$

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

    $$-28=-28$$

81. Put the simplified groups together.

    $$x^2-3\times x-28$$

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

    $$x^2-3\times x-28$$

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

    $$x^2-3\times x-28=0$$

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

    $$(4-7=-3,4\times \left(-7\right)=-28)$$

85. Rewrite the middle term using those two parts.

    $$x^2+4\times x-7\times x-28=0$$

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

    $$x\times \left(x+4\right)-7\times \left(x+4\right)=0$$

87. The common bracket can now be factored out.

    $$\left(x-7\right)\times \left(x+4\right)=0$$

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

    $$(x-7=0,x+4=0)$$

89. Take this factor equal to zero.

    $$x-7=0$$

90. Subtract the constant from both sides.

    $$x-7+7=0+7$$

91. Simplify both sides.

    $$x=7$$

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

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

93. Reduce the fraction to obtain this root.

    $$x=7$$

94. Take this factor equal to zero.

    $$x+4=0$$

95. Subtract the constant from both sides.

    $$x+4-4=0-4$$

96. Simplify both sides.

    $$x=-4$$

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

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

98. Reduce the fraction to obtain this root.

    $$x=-4$$

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

    $$28=28$$

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

     $$28=28$$

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

     $$28=28$$

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

     $$28=28$$

103. Collect every valid root from every substitution branch.

     $$\{-4,-2,5,7\}$$

Answer

The final answer is shown below.