Mathematics 10 — Exercise 2.1 · Q3 (v)

Full question

Solve the following quadratic equation by completing square method: 1/(1+x)-1/(3-x) = 6/35. where x ≠ −1, 3.

Step-by-step solution

1. A denominator cannot equal zero. Keep the original excluded values throughout the calculation.

   $$35\times x^2-70\times x-105\ne 0$$

2. Multiply BOTH sides by the common denominator to remove the fractions.

   $$\left(\frac{1}{x+1}-\frac{1}{3-x}\right)\times \left(35\times x^2-70\times x-105\right)=\frac{210\times x^2-420\times x-630}{35}$$

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

   $$6\times x^2-82\times x+52=0$$

4. Calculate the normalized linear coefficient, then halve it.

   $$-\frac{82}{2\times 6}$$

5. Calculate this operation before combining the remaining terms.

   $$2\times 6=12$$

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

   $$\frac{\left(-82\right)\times 1}{1\times 12}$$

7. Calculate these two products separately.

   $$-\frac{82}{12}$$

8. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

   $$\mathrm{GCD}=2$$

9. Show the two divisions explicitly.

   $$\frac{-\frac{82}{2}}{\frac{12}{2}}$$

10. Evaluate the numerator division.

    $$-\frac{82}{2}=-41$$

11. Evaluate the denominator division.

    $$\frac{12}{2}=6$$

12. Write the reduced fraction.

    $$-\frac{41}{6}$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-82\right)\times \frac{1}{12}=-\frac{41}{6}$$

14. Square this half-coefficient and compute the new right-hand side separately.

    $${\left(-\frac{41}{6}\right)}^2$$

15. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{41}{6}\right)}^2=\frac{1681}{36}$$

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

    $$\frac{\left(-52\right)\times 1}{1\times 6}$$

17. Calculate these two products separately.

    $$-\frac{52}{6}$$

18. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

19. Show the two divisions explicitly.

    $$\frac{-\frac{52}{2}}{\frac{6}{2}}$$

20. Evaluate the numerator division.

    $$-\frac{52}{2}=-26$$

21. Evaluate the denominator division.

    $$\frac{6}{2}=3$$

22. Write the reduced fraction.

    $$-\frac{26}{3}$$

23. Calculate this operation before combining the remaining terms.

    $$\left(-52\right)\times \frac{1}{6}=-\frac{26}{3}$$

24. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{41}{6}\right)}^2=\frac{1681}{36}$$

25. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=36$$

26. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{26}{3}=\frac{\left(-26\right)\times 12}{3\times 12}$$

27. Calculate the converted numerator; the denominator is now common.

    $$-\frac{312}{36}$$

28. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{1681}{36}=\frac{1681\times 1}{36\times 1}$$

29. Calculate the converted numerator; the denominator is now common.

    $$\frac{1681}{36}$$

30. Keep the common denominator and add the signed numerators.

    $$\frac{-312+1681}{36}=\frac{1369}{36}$$

31. Calculate this operation before combining the remaining terms.

    $$-\frac{26}{3}+\frac{1681}{36}=\frac{1369}{36}$$

32. Divide by the coefficient of the squared term and move the constant to the other side.

    $$x^2-\frac{41\times x}{3}=-\frac{26}{3}$$

33. Half the linear coefficient is b/(2a). Add its SQUARE to both sides to keep the equation balanced.

    $$x^2-\frac{41\times x}{3}+{\left(-\frac{41}{6}\right)}^2=-\frac{26}{3}+{\left(-\frac{41}{6}\right)}^2$$

34. The left side is a perfect square. Add the fractions on the right.

    $${\left(x-\frac{41}{6}\right)}^2=\frac{1369}{36}$$

35. Take both square-root signs. Each equation gives a possible root.

    $$(x-\frac{41}{6}=\frac{37}{6},x-\frac{41}{6}=-\frac{37}{6})$$

36. Subtract the half-coefficient from both sides to isolate the variable.

    $$(x=13,x=\frac{2}{3})$$

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

    $$x=1\times \sqrt{\frac{1369}{36}}+\frac{41}{6}$$

38. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{1369}{36}}=\frac{37}{6}$$

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

    $$\frac{1\times 37}{1\times 6}$$

40. Calculate these two products separately.

    $$\frac{37}{6}$$

41. Calculate this operation before combining the remaining terms.

    $$1\times \frac{37}{6}=\frac{37}{6}$$

42. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=6$$

43. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{37}{6}=\frac{37\times 1}{6\times 1}$$

44. Calculate the converted numerator; the denominator is now common.

    $$\frac{37}{6}$$

45. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{41}{6}=\frac{41\times 1}{6\times 1}$$

46. Calculate the converted numerator; the denominator is now common.

    $$\frac{41}{6}$$

47. Keep the common denominator and add the signed numerators.

    $$\frac{37+41}{6}=\frac{78}{6}$$

48. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=6$$

49. Show the two divisions explicitly.

    $$\frac{\frac{78}{6}}{\frac{6}{6}}$$

50. Evaluate the numerator division.

    $$\frac{78}{6}=13$$

51. Evaluate the denominator division.

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

52. Write the reduced fraction.

    $$13\times \frac{1}{1}$$

53. Calculate this operation before combining the remaining terms.

    $$\frac{37}{6}+\frac{41}{6}=13$$

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

    $$x=-\sqrt{\frac{1369}{36}}+\frac{41}{6}$$

55. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{1369}{36}}=\frac{37}{6}$$

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

    $$\frac{\left(-1\right)\times 37}{1\times 6}$$

57. Calculate these two products separately.

    $$-\frac{37}{6}$$

58. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{37}{6}=-\frac{37}{6}$$

59. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=6$$

60. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{37}{6}=\frac{\left(-37\right)\times 1}{6\times 1}$$

61. Calculate the converted numerator; the denominator is now common.

    $$-\frac{37}{6}$$

62. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{41}{6}=\frac{41\times 1}{6\times 1}$$

63. Calculate the converted numerator; the denominator is now common.

    $$\frac{41}{6}$$

64. Keep the common denominator and add the signed numerators.

    $$\frac{-37+41}{6}=\frac{4}{6}$$

65. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

66. Show the two divisions explicitly.

    $$\frac{\frac{4}{2}}{\frac{6}{2}}$$

67. Evaluate the numerator division.

    $$\frac{4}{2}=2$$

68. Evaluate the denominator division.

    $$\frac{6}{2}=3$$

69. Write the reduced fraction.

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

70. Calculate this operation before combining the remaining terms.

    $$-\frac{37}{6}+\frac{41}{6}=\frac{2}{3}$$

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

    $$\frac{6}{35}=\frac{6}{35}$$

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

    $$\frac{6}{35}=\frac{6}{35}$$

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

    $$\frac{6}{35}=\frac{6}{35}$$

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

    $$\frac{6}{35}=\frac{6}{35}$$

Answer

The final answer is shown below.