Mathematics 10 — Exercise 2.1 · Q3 (vi)

Full question

Solve the following quadratic equation by completing square method: (3 × x-1)/(4 × x+7) = 1-6/(x+7). where x ≠ −7/4, −7.

Step-by-step solution

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

   $$4\times x^2+35\times x+49\ne 0$$

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

   $$\left(\frac{3\times x-1}{4\times x+7}\right)\times \left(4\times x^2+35\times x+49\right)=\left(1-\frac{6}{x+7}\right)\times \left(4\times x^2+35\times x+49\right)$$

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

   $$x^2-9\times x+14=0$$

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

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

5. Calculate this operation before combining the remaining terms.

   $$2\times 1=2$$

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

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

7. Calculate these two products separately.

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

8. Calculate this operation before combining the remaining terms.

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

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

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

10. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{9}{2}\right)}^2=\frac{81}{4}$$

11. Calculate this operation before combining the remaining terms.

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

12. Calculate this operation before combining the remaining terms.

    $$\left(-14\right)\times 1=-14$$

13. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{9}{2}\right)}^2=\frac{81}{4}$$

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

    $$\mathrm{LCD}=4$$

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

    $$-14\times \frac{1}{1}=\frac{\left(-14\right)\times 4}{1\times 4}$$

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

    $$-\frac{56}{4}$$

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

    $$\frac{81}{4}=\frac{81\times 1}{4\times 1}$$

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

    $$\frac{81}{4}$$

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

    $$\frac{-56+81}{4}=\frac{25}{4}$$

20. Calculate this operation before combining the remaining terms.

    $$-14+\frac{81}{4}=\frac{25}{4}$$

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

    $$x^2-9\times x=-14$$

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

    $$x^2-9\times x+{\left(-\frac{9}{2}\right)}^2=-14+{\left(-\frac{9}{2}\right)}^2$$

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

    $${\left(x-\frac{9}{2}\right)}^2=\frac{25}{4}$$

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

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

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

    $$(x=7,x=2)$$

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

    $$x=1\times \sqrt{\frac{25}{4}}+\frac{9}{2}$$

27. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{25}{4}}=\frac{5}{2}$$

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

    $$\frac{1\times 5}{1\times 2}$$

29. Calculate these two products separately.

    $$\frac{5}{2}$$

30. Calculate this operation before combining the remaining terms.

    $$1\times \frac{5}{2}=\frac{5}{2}$$

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

    $$\mathrm{LCD}=2$$

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

    $$\frac{5}{2}=\frac{5\times 1}{2\times 1}$$

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

    $$\frac{5}{2}$$

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

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

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

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

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

    $$\frac{5+9}{2}=\frac{14}{2}$$

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

    $$\mathrm{GCD}=2$$

38. Show the two divisions explicitly.

    $$\frac{\frac{14}{2}}{\frac{2}{2}}$$

39. Evaluate the numerator division.

    $$\frac{14}{2}=7$$

40. Evaluate the denominator division.

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

41. Write the reduced fraction.

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

42. Calculate this operation before combining the remaining terms.

    $$\frac{5}{2}+\frac{9}{2}=7$$

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

    $$x=-\sqrt{\frac{25}{4}}+\frac{9}{2}$$

44. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{25}{4}}=\frac{5}{2}$$

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

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

46. Calculate these two products separately.

    $$-\frac{5}{2}$$

47. Calculate this operation before combining the remaining terms.

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

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

    $$\mathrm{LCD}=2$$

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

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

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

    $$-\frac{5}{2}$$

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

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

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

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

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

    $$\frac{-5+9}{2}=\frac{4}{2}$$

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

    $$\mathrm{GCD}=2$$

55. Show the two divisions explicitly.

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

56. Evaluate the numerator division.

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

57. Evaluate the denominator division.

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

58. Write the reduced fraction.

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

59. Calculate this operation before combining the remaining terms.

    $$-\frac{5}{2}+\frac{9}{2}=2$$

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

    $$\frac{4}{7}=\frac{4}{7}$$

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

    $$\frac{4}{7}=\frac{4}{7}$$

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

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

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

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

Answer

The final answer is shown below.