Mathematics 10 — Exercise Review · Q4 (vi)

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.

   $$\left(\left(x-7\right)\times \left(x+9\right)\right)\times \left(\left(x-3\right)\times \left(x+5\right)\right)$$

2. Expand the first pair.

   $$\left(x-7\right)\times \left(x+9\right)=x^2+2\times x-63$$

3. Expand the second pair.

   $$\left(x-3\right)\times \left(x+5\right)=x^2+2\times x-15$$

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

   $$t=x^2+2\times x$$

5. The product is now a quadratic in t.

   $$\left(t-63\right)\times \left(t-15\right)=385$$

6. After substitution the quadratic equation is:

   $$t^2-78\times t+560=0$$

7. Substitute the function expressions, keeping the brackets.

   $$t^2-78\times t+560-0$$

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

   $$t^2-78\times t+560$$

9. Write EVERY distributed product before collecting terms.

   $$560+t^2-78\times t-0$$

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

    $$560=560$$

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

    $$t^2=t^2$$

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

    $$-78\times t=-78\times t$$

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.

    $$560+0=560$$

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.

    $$-78=-78$$

17. Put the simplified groups together.

    $$t^2-78\times t+560$$

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

    $$t^2-78\times t+560$$

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

    $$t^2-78\times t+560=0$$

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

    $$(-8-70=-78,\left(-8\right)\times \left(-70\right)=560)$$

21. Rewrite the middle term using those two parts.

    $$t^2-8\times t-70\times t+560=0$$

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

    $$t\times \left(t-8\right)-70\times \left(t-8\right)=0$$

23. The common bracket can now be factored out.

    $$\left(t-70\right)\times \left(t-8\right)=0$$

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

    $$(t-70=0,t-8=0)$$

25. Take this factor equal to zero.

    $$t-70=0$$

26. Subtract the constant from both sides.

    $$t-70+70=0+70$$

27. Simplify both sides.

    $$t=70$$

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

    $$t=70\times \frac{1}{1}$$

29. Reduce the fraction to obtain this root.

    $$t=70$$

30. Take this factor equal to zero.

    $$t-8=0$$

31. Subtract the constant from both sides.

    $$t-8+8=0+8$$

32. Simplify both sides.

    $$t=8$$

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

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

34. Reduce the fraction to obtain this root.

    $$t=8$$

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.

    $$x^2+2\times x=8$$

40. Substitute the function expressions, keeping the brackets.

    $$x^2+2\times x-8$$

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

    $$x^2+2\times x-8$$

42. Write EVERY distributed product before collecting terms.

    $$x^2+2\times x-8$$

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

    $$x^2=x^2$$

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

    $$2\times x=2\times x$$

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

    $$\left(-1\right)\times 8=-8$$

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.

    $$2=2$$

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

    $$-8=-8$$

49. Put the simplified groups together.

    $$x^2+2\times x-8$$

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

    $$x^2+2\times x-8$$

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

    $$x^2+2\times x-8=0$$

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

    $$(4-2=2,4\times \left(-2\right)=-8)$$

53. Rewrite the middle term using those two parts.

    $$x^2+4\times x-2\times x-8=0$$

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

    $$x\times \left(x+4\right)-2\times \left(x+4\right)=0$$

55. The common bracket can now be factored out.

    $$\left(x-2\right)\times \left(x+4\right)=0$$

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

    $$(x-2=0,x+4=0)$$

57. Take this factor equal to zero.

    $$x-2=0$$

58. Subtract the constant from both sides.

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

59. Simplify both sides.

    $$x=2$$

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

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

61. Reduce the fraction to obtain this root.

    $$x=2$$

62. Take this factor equal to zero.

    $$x+4=0$$

63. Subtract the constant from both sides.

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

64. Simplify both sides.

    $$x=-4$$

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

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

66. Reduce the fraction to obtain this root.

    $$x=-4$$

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

    $$8=8$$

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

    $$8=8$$

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

    $$8=8$$

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

    $$8=8$$

71. Restore x²+sx=t for this root.

    $$x^2+2\times x=70$$

72. Substitute the function expressions, keeping the brackets.

    $$x^2+2\times x-70$$

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

    $$x^2+2\times x-70$$

74. Write EVERY distributed product before collecting terms.

    $$x^2+2\times x-70$$

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

    $$x^2=x^2$$

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

    $$2\times x=2\times x$$

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

    $$\left(-1\right)\times 70=-70$$

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.

    $$2=2$$

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

    $$-70=-70$$

81. Put the simplified groups together.

    $$x^2+2\times x-70$$

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

    $$x^2+2\times x-70$$

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

    $$x^2+2\times x-70=0$$

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

85. Calculate this operation before combining the remaining terms.

    $$4\times \left(-70\right)=-280$$

86. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-280\right)=280$$

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

    $$2^2-4\times \left(-70\right)$$

88. Calculate this operation before combining the remaining terms.

    $$2^2=4$$

89. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times \left(-70\right)=280$$

90. Calculate this operation before combining the remaining terms.

    $$4+280=284$$

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

    $$(a=1,b=2,c=-70)$$

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

    $$D=2^2-4\times \left(-70\right)$$

93. Simplify the discriminant.

    $$D=284$$

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

    $$(x=\frac{-2+2\times \sqrt{71}}{2},x=\frac{\left(-2\right)\times \sqrt{71}-2}{2})$$

95. Simplify each numerator and divide by 2a.

    $$\{-1+\sqrt{71},\left(-1\right)\times \sqrt{71}-1\}$$

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

    $$x=\frac{-2+1\times \sqrt{284}}{2\times 1}$$

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

    $$284=4\times 71$$

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

    $$\sqrt{284}=2\times \sqrt{71}$$

99. Calculate this operation before combining the remaining terms.

    $$\sqrt{284}=2\times \sqrt{71}$$

100. Calculate this operation before combining the remaining terms.

     $$1\times 2\times \sqrt{71}=2\times \sqrt{71}$$

101. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

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

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

103. Calculate these two products separately.

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

104. Calculate this operation before combining the remaining terms.

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

105. Calculate this operation before combining the remaining terms.

     $$-2+2\times \sqrt{71}\times \frac{1}{2}=-1+\sqrt{71}$$

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

     $$x=\frac{\left(-1\right)\times \sqrt{284}-2}{2\times 1}$$

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

     $$284=4\times 71$$

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

     $$\sqrt{284}=2\times \sqrt{71}$$

109. Calculate this operation before combining the remaining terms.

     $$\sqrt{284}=2\times \sqrt{71}$$

110. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 2\times \sqrt{71}=\left(-2\right)\times \sqrt{71}$$

111. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 2=-2$$

113. Calculate this operation before combining the remaining terms.

     $$-2\times \sqrt{71}-2=\left(-2\right)\times \sqrt{71}-2$$

114. Calculate this operation before combining the remaining terms.

     $$\left(\left(-2\right)\times \sqrt{71}-2\right)\times \frac{1}{2}=\left(-1\right)\times \sqrt{71}-1$$

115. Calculate this operation before combining the remaining terms.

     $${\left(-1+\sqrt{71}\right)}^2={\left(1-\sqrt{71}\right)}^2$$

116. Calculate this operation before combining the remaining terms.

     $$-2+2\times \sqrt{71}+{\left(1-\sqrt{71}\right)}^2=70$$

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

     $$-2+2\times \sqrt{71}+{\left(-1+\sqrt{71}\right)}^2=70$$

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

     $$70=70$$

119. Calculate this operation before combining the remaining terms.

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

120. Calculate this operation before combining the remaining terms.

     $$-\sqrt{71}-1=\left(-1\right)\times \sqrt{71}-1$$

121. Calculate this operation before combining the remaining terms.

     $${\left(\left(-1\right)\times \sqrt{71}-1\right)}^2={\left(1+\sqrt{71}\right)}^2$$

122. Calculate this operation before combining the remaining terms.

     $$-2\times \sqrt{71}-2+{\left(1+\sqrt{71}\right)}^2=70$$

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

     $$\left(-2\right)\times \sqrt{71}-2+{\left(\left(-1\right)\times \sqrt{71}-1\right)}^2=70$$

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

     $$70=70$$

125. Collect every valid root from every substitution branch.

     $$\{-4,2,-1+\sqrt{71},\left(-1\right)\times \sqrt{71}-1\}$$

Answer

The final answer is shown below.