Mathematics 10 — Exercise 5.4 · Q1 (xi)

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.

   $$\left(\left(x-5\right)\times \left(x+2\right)\right)\times \left(\left(x-4\right)\times \left(x+1\right)\right)$$

2. Expand the first pair.

   $$\left(x-4\right)\times \left(x+1\right)=x^2-3\times x-4$$

3. Expand the second pair.

   $$\left(x-5\right)\times \left(x+2\right)=x^2-3\times x-10$$

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.

   $$\left(t-10\right)\times \left(t-4\right)=-8$$

6. After substitution the quadratic equation is:

   $$t^2-14\times t+48=0$$

7. Substitute the function expressions, keeping the brackets.

   $$t^2-14\times t+48-0$$

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

   $$t^2-14\times t+48$$

9. Write EVERY distributed product before collecting terms.

   $$48+t^2-14\times t-0$$

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

    $$48=48$$

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

    $$t^2=t^2$$

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

    $$-14\times t=-14\times t$$

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.

    $$48+0=48$$

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.

    $$-14=-14$$

17. Put the simplified groups together.

    $$t^2-14\times t+48$$

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

    $$t^2-14\times t+48$$

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

    $$t^2-14\times t+48=0$$

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

    $$(-6-8=-14,\left(-6\right)\times \left(-8\right)=48)$$

21. Rewrite the middle term using those two parts.

    $$t^2-6\times t-8\times t+48=0$$

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

    $$t\times \left(t-6\right)-8\times \left(t-6\right)=0$$

23. The common bracket can now be factored out.

    $$\left(t-8\right)\times \left(t-6\right)=0$$

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

    $$(t-8=0,t-6=0)$$

25. Take this factor equal to zero.

    $$t-8=0$$

26. Subtract the constant from both sides.

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

27. Simplify both sides.

    $$t=8$$

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

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

29. Reduce the fraction to obtain this root.

    $$t=8$$

30. Take this factor equal to zero.

    $$t-6=0$$

31. Subtract the constant from both sides.

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

32. Simplify both sides.

    $$t=6$$

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

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

34. Reduce the fraction to obtain this root.

    $$t=6$$

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.

    $$x^2-3\times x=6$$

40. Substitute the function expressions, keeping the brackets.

    $$x^2-3\times x-6$$

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

    $$x^2-3\times x-6$$

42. Write EVERY distributed product before collecting terms.

    $$x^2-3\times x-6$$

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.

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

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.

    $$-6=-6$$

49. Put the simplified groups together.

    $$x^2-3\times x-6$$

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

    $$x^2-3\times x-6$$

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

    $$x^2-3\times x-6=0$$

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

53. Calculate this operation before combining the remaining terms.

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

54. Calculate this operation before combining the remaining terms.

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

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

    $${\left(-3\right)}^2-4\times \left(-6\right)$$

56. Calculate this operation before combining the remaining terms.

    $${\left(-3\right)}^2=9$$

57. Calculate this operation before combining the remaining terms.

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

58. Calculate this operation before combining the remaining terms.

    $$9+24=33$$

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

    $$(a=1,b=-3,c=-6)$$

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

    $$D={\left(-3\right)}^2-4\times \left(-6\right)$$

61. Simplify the discriminant.

    $$D=33$$

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

    $$(x=\frac{3+\sqrt{33}}{2},x=\frac{3-\sqrt{33}}{2})$$

63. Simplify each numerator and divide by 2a.

    $$\{\frac{3}{2}-\frac{1}{2}\times \sqrt{33},\frac{3}{2}+\frac{1}{2}\times \sqrt{33}\}$$

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

    $$x=\frac{3+1\times \sqrt{33}}{2\times 1}$$

65. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{33}=\sqrt{33}$$

66. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

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

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

68. Calculate these two products separately.

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

69. Calculate this operation before combining the remaining terms.

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

70. Calculate this operation before combining the remaining terms.

    $$3+\sqrt{33}\times \frac{1}{2}=\frac{3}{2}+\frac{1}{2}\times \sqrt{33}$$

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

    $$x=\frac{3-\sqrt{33}}{2\times 1}$$

72. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

73. Calculate this operation before combining the remaining terms.

    $$\left(3-\sqrt{33}\right)\times \frac{1}{2}=\frac{3}{2}-\frac{1}{2}\times \sqrt{33}$$

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

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

75. Calculate these two products separately.

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

76. Calculate this operation before combining the remaining terms.

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

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

    $$\frac{9\times 1}{1\times 2}$$

78. Calculate these two products separately.

    $$\frac{9}{2}$$

79. Calculate this operation before combining the remaining terms.

    $$9\times \frac{1}{2}=\frac{9}{2}$$

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

    $$\frac{\left(-1\right)\times 9}{1\times 2}$$

81. Calculate these two products separately.

    $$-\frac{9}{2}$$

82. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{9}{2}=-\frac{9}{2}$$

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

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

84. Calculate these two products separately.

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

85. Calculate this operation before combining the remaining terms.

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

86. Calculate this operation before combining the remaining terms.

    $${\left(\frac{3}{2}+\frac{1}{2}\times \sqrt{33}\right)}^2=\frac{1}{4}\times {\left(3+\sqrt{33}\right)}^2$$

87. Calculate this operation before combining the remaining terms.

    $$-\frac{3}{2}\times \sqrt{33}-\frac{9}{2}+\frac{1}{4}\times {\left(3+\sqrt{33}\right)}^2=6$$

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

    $$\left(-\frac{3}{2}\right)\times \sqrt{33}-\frac{9}{2}+{\left(\frac{3}{2}+\frac{1}{2}\times \sqrt{33}\right)}^2=6$$

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

    $$6=6$$

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

    $$\frac{\left(-9\right)\times 1}{1\times 2}$$

91. Calculate this operation before combining the remaining terms.

    $$\left(-9\right)\times \frac{1}{2}=-\frac{9}{2}$$

92. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{2}\times \sqrt{33}=\left(-\frac{1}{2}\right)\times \sqrt{33}$$

93. Calculate this operation before combining the remaining terms.

    $${\left(\frac{3}{2}-\frac{1}{2}\times \sqrt{33}\right)}^2=\frac{1}{4}\times {\left(3-\sqrt{33}\right)}^2$$

94. Calculate this operation before combining the remaining terms.

    $$-\frac{9}{2}+\frac{1}{4}\times {\left(3-\sqrt{33}\right)}^2+\frac{3}{2}\times \sqrt{33}=6$$

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

    $$-\frac{9}{2}+{\left(\frac{3}{2}-\frac{1}{2}\times \sqrt{33}\right)}^2+\frac{3}{2}\times \sqrt{33}=6$$

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

    $$6=6$$

97. Restore x²+sx=t for this root.

    $$x^2-3\times x=8$$

98. Substitute the function expressions, keeping the brackets.

    $$x^2-3\times x-8$$

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

    $$x^2-3\times x-8$$

100. Write EVERY distributed product before collecting terms.

     $$x^2-3\times x-8$$

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

     $$x^2=x^2$$

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

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

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

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

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

     $$1=1$$

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

     $$-3=-3$$

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

     $$-8=-8$$

107. Put the simplified groups together.

     $$x^2-3\times x-8$$

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

     $$x^2-3\times x-8$$

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

     $$x^2-3\times x-8=0$$

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

111. Calculate this operation before combining the remaining terms.

     $$4\times \left(-8\right)=-32$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \left(-32\right)=32$$

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

     $${\left(-3\right)}^2-4\times \left(-8\right)$$

114. Calculate this operation before combining the remaining terms.

     $${\left(-3\right)}^2=9$$

115. Calculate this operation before combining the remaining terms.

     $$\left(-4\right)\times 1\times \left(-8\right)=32$$

116. Calculate this operation before combining the remaining terms.

     $$9+32=41$$

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

     $$(a=1,b=-3,c=-8)$$

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

     $$D={\left(-3\right)}^2-4\times \left(-8\right)$$

119. Simplify the discriminant.

     $$D=41$$

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

     $$(x=\frac{3+\sqrt{41}}{2},x=\frac{3-\sqrt{41}}{2})$$

121. Simplify each numerator and divide by 2a.

     $$\{\frac{3}{2}-\frac{1}{2}\times \sqrt{41},\frac{3}{2}+\frac{1}{2}\times \sqrt{41}\}$$

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

     $$x=\frac{3+1\times \sqrt{41}}{2\times 1}$$

123. Calculate this operation before combining the remaining terms.

     $$1\times \sqrt{41}=\sqrt{41}$$

124. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

125. Calculate this operation before combining the remaining terms.

     $$3+\sqrt{41}\times \frac{1}{2}=\frac{3}{2}+\frac{1}{2}\times \sqrt{41}$$

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

     $$x=\frac{3-\sqrt{41}}{2\times 1}$$

127. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

128. Calculate this operation before combining the remaining terms.

     $$\left(3-\sqrt{41}\right)\times \frac{1}{2}=\frac{3}{2}-\frac{1}{2}\times \sqrt{41}$$

129. Calculate this operation before combining the remaining terms.

     $${\left(\frac{3}{2}+\frac{1}{2}\times \sqrt{41}\right)}^2=\frac{1}{4}\times {\left(3+\sqrt{41}\right)}^2$$

130. Calculate this operation before combining the remaining terms.

     $$-\frac{3}{2}\times \sqrt{41}-\frac{9}{2}+\frac{1}{4}\times {\left(3+\sqrt{41}\right)}^2=8$$

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

     $$\left(-\frac{3}{2}\right)\times \sqrt{41}-\frac{9}{2}+{\left(\frac{3}{2}+\frac{1}{2}\times \sqrt{41}\right)}^2=8$$

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

     $$8=8$$

133. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{1}{2}\times \sqrt{41}=\left(-\frac{1}{2}\right)\times \sqrt{41}$$

134. Calculate this operation before combining the remaining terms.

     $${\left(\frac{3}{2}-\frac{1}{2}\times \sqrt{41}\right)}^2=\frac{1}{4}\times {\left(3-\sqrt{41}\right)}^2$$

135. Calculate this operation before combining the remaining terms.

     $$-\frac{9}{2}+\frac{1}{4}\times {\left(3-\sqrt{41}\right)}^2+\frac{3}{2}\times \sqrt{41}=8$$

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

     $$-\frac{9}{2}+{\left(\frac{3}{2}-\frac{1}{2}\times \sqrt{41}\right)}^2+\frac{3}{2}\times \sqrt{41}=8$$

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

     $$8=8$$

138. Collect every valid root from every substitution branch.

     $$\{\frac{3}{2}-\frac{1}{2}\times \sqrt{33},\frac{3}{2}+\frac{1}{2}\times \sqrt{33},\frac{3}{2}-\frac{1}{2}\times \sqrt{41},\frac{3}{2}+\frac{1}{2}\times \sqrt{41}\}$$

Answer

The final answer is shown below.