Mathematics 10 — Exercise 5.4 · Q1 (xii)

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.

   $$\left(\left(x+3\right)\times \left(x+6\right)\right)\times \left(\left(x+4\right)\times \left(x+5\right)\right)$$

2. Expand the first pair.

   $$\left(x+3\right)\times \left(x+6\right)=x^2+9\times x+18$$

3. Expand the second pair.

   $$\left(x+4\right)\times \left(x+5\right)=x^2+9\times x+20$$

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

   $$t=x^2+9\times x$$

5. The product is now a quadratic in t.

   $$\left(t+18\right)\times \left(t+20\right)=-1$$

6. After substitution the quadratic equation is:

   $$t^2+38\times t+361=0$$

7. Substitute the function expressions, keeping the brackets.

   $$t^2+38\times t+361-0$$

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

   $$t^2+38\times t+361$$

9. Write EVERY distributed product before collecting terms.

   $$361+t^2+38\times t-0$$

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

    $$361=361$$

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.

    $$361+0=361$$

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.

    $$t^2+38\times t+361$$

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

    $$t^2+38\times t+361$$

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

    $$t^2+38\times t+361=0$$

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

    $$(19+19=38,19\times 19=361)$$

21. Rewrite the middle term using those two parts.

    $$t^2+19\times t+19\times t+361=0$$

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

    $$t\times \left(t+19\right)+19\times \left(t+19\right)=0$$

23. The common bracket can now be factored out.

    $$\left(t+19\right)\left(t+19\right)=0$$

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

    $$(t+19=0,t+19=0)$$

25. Take this factor equal to zero.

    $$t+19=0$$

26. Subtract the constant from both sides.

    $$t+19-19=0-19$$

27. Simplify both sides.

    $$t=-19$$

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

    $$t=-19\times \frac{1}{1}$$

29. Reduce the fraction to obtain this root.

    $$t=-19$$

30. Take this factor equal to zero.

    $$t+19=0$$

31. Subtract the constant from both sides.

    $$t+19-19=0-19$$

32. Simplify both sides.

    $$t=-19$$

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

    $$t=-19\times \frac{1}{1}$$

34. Reduce the fraction to obtain this root.

    $$t=-19$$

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. Restore x²+sx=t for this root.

    $$x^2+9\times x=-19$$

38. Substitute the function expressions, keeping the brackets.

    $$x^2+9\times x--19$$

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

    $$x^2+9\times x+19$$

40. Write EVERY distributed product before collecting terms.

    $$x^2+9\times x--19$$

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

    $$x^2=x^2$$

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

    $$9\times x=9\times x$$

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

    $$\left(-1\right)\times \left(-19\right)=19$$

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

    $$1=1$$

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

    $$9=9$$

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

    $$19=19$$

47. Put the simplified groups together.

    $$x^2+9\times x+19$$

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

    $$x^2+9\times x+19$$

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

    $$x^2+9\times x+19=0$$

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

51. Calculate this operation before combining the remaining terms.

    $$4\times 19=76$$

52. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 76=-76$$

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

    $$9^2-4\times 19$$

54. Calculate this operation before combining the remaining terms.

    $$9^2=81$$

55. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times 19=-76$$

56. Calculate this operation before combining the remaining terms.

    $$81-76=5$$

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

    $$(a=1,b=9,c=19)$$

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

    $$D=9^2-4\times 19$$

59. Simplify the discriminant.

    $$D=5$$

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

    $$(x=\frac{-9+\sqrt{5}}{2},x=\frac{-9-\sqrt{5}}{2})$$

61. Simplify each numerator and divide by 2a.

    $$\{-\frac{9}{2}-\frac{1}{2}\times \sqrt{5},-\frac{9}{2}+\frac{1}{2}\times \sqrt{5}\}$$

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

    $$x=\frac{-9+1\times \sqrt{5}}{2\times 1}$$

63. Calculate this operation before combining the remaining terms.

    $$1\times \sqrt{5}=\sqrt{5}$$

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.

    $$\left(-9+\sqrt{5}\right)\times \frac{1}{2}=-\frac{9}{2}+\frac{1}{2}\times \sqrt{5}$$

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

    $$x=\frac{-9-\sqrt{5}}{2\times 1}$$

70. Calculate this operation before combining the remaining terms.

    $$2\times 1=2$$

71. Calculate this operation before combining the remaining terms.

    $$\left(-9-\sqrt{5}\right)\times \frac{1}{2}=-\frac{9}{2}-\frac{1}{2}\times \sqrt{5}$$

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

    $$\frac{\left(-81\right)\times 1}{1\times 2}$$

73. Calculate these two products separately.

    $$-\frac{81}{2}$$

74. Calculate this operation before combining the remaining terms.

    $$\left(-81\right)\times \frac{1}{2}=-\frac{81}{2}$$

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

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

76. Calculate these two products separately.

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

77. Calculate this operation before combining the remaining terms.

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

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

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

79. Calculate these two products separately.

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

80. Calculate this operation before combining the remaining terms.

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

81. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{9}{2}+\frac{1}{2}\times \sqrt{5}\right)}^2=\frac{1}{4}\times {\left(9-\sqrt{5}\right)}^2$$

82. Calculate this operation before combining the remaining terms.

    $$-\frac{81}{2}+\frac{9}{2}\times \sqrt{5}+\frac{1}{4}\times {\left(9-\sqrt{5}\right)}^2=-19$$

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

    $$-\frac{81}{2}+\frac{9}{2}\times \sqrt{5}+{\left(-\frac{9}{2}+\frac{1}{2}\times \sqrt{5}\right)}^2=-19$$

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

    $$-19=-19$$

85. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{9}{2}\times \sqrt{5}=\left(-\frac{9}{2}\right)\times \sqrt{5}$$

86. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{1}{2}\times \sqrt{5}=\left(-\frac{1}{2}\right)\times \sqrt{5}$$

87. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{9}{2}-\frac{1}{2}\times \sqrt{5}\right)}^2=\frac{1}{4}\times {\left(\sqrt{5}+9\right)}^2$$

88. Calculate this operation before combining the remaining terms.

    $$-\frac{81}{2}-\frac{9}{2}\times \sqrt{5}+\frac{1}{4}\times {\left(\sqrt{5}+9\right)}^2=-19$$

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

    $$-\frac{81}{2}-\frac{9}{2}\times \sqrt{5}+{\left(-\frac{9}{2}-\frac{1}{2}\times \sqrt{5}\right)}^2=-19$$

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

    $$-19=-19$$

91. Collect every valid root from every substitution branch.

    $$\{-\frac{9}{2}-\frac{1}{2}\times \sqrt{5},-\frac{9}{2}+\frac{1}{2}\times \sqrt{5}\}$$

Answer

The final answer is shown below.