Mathematics 10 — Exercise 2.1 · Q3 (iv)

Full question

Solve the following quadratic equation by completing square method: (x+3)/(2 × x-7) = (2 × x-1)/(x-3). where x ≠ 7/2, 3.

Step-by-step solution

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

   $$2\times x^2-13\times x+21\ne 0$$

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

   $$\left(\frac{x+3}{2\times x-7}\right)\times \left(2\times x^2-13\times x+21\right)=\left(\frac{2\times x-1}{x-3}\right)\times \left(2\times x^2-13\times x+21\right)$$

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

   $$3\times x^2-16\times x+16=0$$

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

   $$-\frac{16}{2\times 3}$$

5. Calculate this operation before combining the remaining terms.

   $$2\times 3=6$$

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

   $$\frac{\left(-16\right)\times 1}{1\times 6}$$

7. Calculate these two products separately.

   $$-\frac{16}{6}$$

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

   $$\mathrm{GCD}=2$$

9. Show the two divisions explicitly.

   $$\frac{-\frac{16}{2}}{\frac{6}{2}}$$

10. Evaluate the numerator division.

    $$-\frac{16}{2}=-8$$

11. Evaluate the denominator division.

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

12. Write the reduced fraction.

    $$-\frac{8}{3}$$

13. Calculate this operation before combining the remaining terms.

    $$\left(-16\right)\times \frac{1}{6}=-\frac{8}{3}$$

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

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

15. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{8}{3}\right)}^2=\frac{64}{9}$$

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

    $$\frac{\left(-16\right)\times 1}{1\times 3}$$

17. Calculate these two products separately.

    $$-\frac{16}{3}$$

18. Calculate this operation before combining the remaining terms.

    $$\left(-16\right)\times \frac{1}{3}=-\frac{16}{3}$$

19. Calculate this operation before combining the remaining terms.

    $${\left(-\frac{8}{3}\right)}^2=\frac{64}{9}$$

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

    $$\mathrm{LCD}=9$$

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

    $$-\frac{16}{3}=\frac{\left(-16\right)\times 3}{3\times 3}$$

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

    $$-\frac{48}{9}$$

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

    $$\frac{64}{9}=\frac{64\times 1}{9\times 1}$$

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

    $$\frac{64}{9}$$

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

    $$\frac{-48+64}{9}=\frac{16}{9}$$

26. Calculate this operation before combining the remaining terms.

    $$-\frac{16}{3}+\frac{64}{9}=\frac{16}{9}$$

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

    $$x^2-\frac{16\times x}{3}=-\frac{16}{3}$$

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

    $$x^2-\frac{16\times x}{3}+{\left(-\frac{8}{3}\right)}^2=-\frac{16}{3}+{\left(-\frac{8}{3}\right)}^2$$

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

    $${\left(x-\frac{8}{3}\right)}^2=\frac{16}{9}$$

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

    $$(x-\frac{8}{3}=\frac{4}{3},x-\frac{8}{3}=-\frac{4}{3})$$

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

    $$(x=4,x=\frac{4}{3})$$

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

    $$x=1\times \sqrt{\frac{16}{9}}+\frac{8}{3}$$

33. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{16}{9}}=\frac{4}{3}$$

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

    $$\frac{1\times 4}{1\times 3}$$

35. Calculate these two products separately.

    $$\frac{4}{3}$$

36. Calculate this operation before combining the remaining terms.

    $$1\times \frac{4}{3}=\frac{4}{3}$$

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

    $$\mathrm{LCD}=3$$

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

    $$\frac{4}{3}=\frac{4\times 1}{3\times 1}$$

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

    $$\frac{4}{3}$$

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

    $$\frac{8}{3}=\frac{8\times 1}{3\times 1}$$

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

    $$\frac{8}{3}$$

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

    $$\frac{4+8}{3}=\frac{12}{3}$$

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

    $$\mathrm{GCD}=3$$

44. Show the two divisions explicitly.

    $$\frac{\frac{12}{3}}{\frac{3}{3}}$$

45. Evaluate the numerator division.

    $$\frac{12}{3}=4$$

46. Evaluate the denominator division.

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

47. Write the reduced fraction.

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

48. Calculate this operation before combining the remaining terms.

    $$\frac{4}{3}+\frac{8}{3}=4$$

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

    $$x=-\sqrt{\frac{16}{9}}+\frac{8}{3}$$

50. Calculate this operation before combining the remaining terms.

    $$\sqrt{\frac{16}{9}}=\frac{4}{3}$$

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

    $$\frac{\left(-1\right)\times 4}{1\times 3}$$

52. Calculate these two products separately.

    $$-\frac{4}{3}$$

53. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \frac{4}{3}=-\frac{4}{3}$$

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

    $$\mathrm{LCD}=3$$

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

    $$-\frac{4}{3}=\frac{\left(-4\right)\times 1}{3\times 1}$$

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

    $$-\frac{4}{3}$$

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

    $$\frac{8}{3}=\frac{8\times 1}{3\times 1}$$

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

    $$\frac{8}{3}$$

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

    $$\frac{-4+8}{3}=\frac{4}{3}$$

60. Calculate this operation before combining the remaining terms.

    $$-\frac{4}{3}+\frac{8}{3}=\frac{4}{3}$$

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

    $$7=7$$

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

    $$7=7$$

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

    $$-1=-1$$

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

    $$-1=-1$$

Answer

The final answer is shown below.