Mathematics 10 — Exercise 1.2 · Q5

Full question

If (1+i)²/(2−i) = x + iy, find x and y.

Step-by-step solution

1. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

   $$i+1+i\times \left(i+1\right)$$

2. Distribute each of those products. Keep all four multiplications visible.

   $$1\times 1+1\times i+1\times i+1\times ii$$

3. Real × real: multiply the numerical coefficients.

   $$1\times 1=1$$

4. Real × imaginary: multiply the numerical coefficients.

   $$1\times i=i$$

5. Imaginary × real: multiply the numerical coefficients.

   $$1\times i=i$$

6. Imaginary × imaginary: multiply the numerical coefficients.

   $$1\times ii=i^2$$

7. Substitute these four products into the expansion.

   $$1+i+i+i^2$$

8. In the last term replace i² by −1.

   $$i^2=1\times \left(-1\right)$$

9. Multiply its coefficient by −1.

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

10. Rewrite the expansion after this sign calculation.

    $$1+i+i-1$$

11. Group the real terms and factor i out of the imaginary terms.

    $$1-1+i\times \left(1+1\right)$$

12. Calculate the sum of the real terms.

    $$1-1=0$$

13. Calculate the sum of the imaginary coefficients.

    $$1+1=2$$

14. Insert both calculated sums in a + bi form.

    $$0+2\times i$$

15. Start with the original quotient. For an inverse, the numerator is 1.

    $$\frac{2\times i}{2-i}$$

16. Find the conjugate of the denominator: keep its real part and reverse the imaginary sign.

    $$i+2$$

17. Multiply by the conjugate over itself; this fraction equals 1.

    $$\left(\frac{2\times i}{2-i}\right)\times \left(\frac{i+2}{i+2}\right)$$

18. Multiply numerators together and denominators together. Keep their brackets intact.

    $$\frac{\left(2\times i\right)\times \left(i+2\right)}{\left(2-i\right)\times \left(i+2\right)}$$

19. Calculate the numerator first.

20. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$0\times \left(i+2\right)+2\times i\times \left(i+2\right)$$

21. Distribute each of those products. Keep all four multiplications visible.

    $$0\times 2+0\times 1\times i+2\times 2\times i+2\times 1\times ii$$

22. Real × real: multiply the numerical coefficients.

    $$0\times 2=0$$

23. Real × imaginary: multiply the numerical coefficients.

    $$0\times 1\times i=0\times i$$

24. Imaginary × real: multiply the numerical coefficients.

    $$2\times 2\times i=4\times i$$

25. Imaginary × imaginary: multiply the numerical coefficients.

    $$2\times 1\times ii=2\times i^2$$

26. Substitute these four products into the expansion.

    $$0+0\times i+4\times i+2\times i^2$$

27. In the last term replace i² by −1.

    $$2\times i^2=2\times \left(-1\right)$$

28. Multiply its coefficient by −1.

    $$2\times \left(-1\right)=-2$$

29. Rewrite the expansion after this sign calculation.

    $$0+0\times i+4\times i-2$$

30. Group the real terms and factor i out of the imaginary terms.

    $$0-2+i\times \left(0+4\right)$$

31. Calculate the sum of the real terms.

    $$0-2=-2$$

32. Calculate the sum of the imaginary coefficients.

    $$0+4=4$$

33. Insert both calculated sums in a + bi form.

    $$4\times i-2$$

34. Now calculate the denominator independently.

35. Distribute the first bracket: multiply its real term by the whole second bracket, then its imaginary term by that bracket.

    $$-i\times \left(i+2\right)+2\times \left(i+2\right)$$

36. Distribute each of those products. Keep all four multiplications visible.

    $$2\times 2+2\times 1\times i-2\times i-1\times ii$$

37. Real × real: multiply the numerical coefficients.

    $$2\times 2=4$$

38. Real × imaginary: multiply the numerical coefficients.

    $$2\times 1\times i=2\times i$$

39. Imaginary × real: multiply the numerical coefficients.

    $$-2\times i=-2\times i$$

40. Imaginary × imaginary: multiply the numerical coefficients.

    $$-1\times ii=-i^2$$

41. Substitute these four products into the expansion.

    $$4+2\times i-2\times i-i^2$$

42. In the last term replace i² by −1.

    $$-i^2=\left(-1\right)\times \left(-1\right)$$

43. Multiply its coefficient by −1.

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

44. Rewrite the expansion after this sign calculation.

    $$4+2\times i-2\times i+1$$

45. Group the real terms and factor i out of the imaginary terms.

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

46. Calculate the sum of the real terms.

    $$4+1=5$$

47. Calculate the sum of the imaginary coefficients.

    $$2-2=0$$

48. Insert both calculated sums in a + bi form.

    $$0\times i+5$$

49. The imaginary coefficient in the denominator is zero, so the denominator is real.

    $$5=5$$

50. Put the calculated numerator above the calculated denominator.

    $$\frac{4\times i-2}{5}$$

51. Separate the real term and imaginary term into two fractions with the same denominator.

    $$-\frac{2}{5}+\frac{4\times i}{5}$$

52. Move i outside its numerical fraction.

    $$-\frac{2}{5}+\left(\frac{4}{5}\right)\times i$$

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

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

54. Calculate these two products separately.

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

55. Calculate this operation before combining the remaining terms.

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

56. Simplify the real coefficient fraction.

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

57. Simplify the imaginary coefficient fraction.

    $$\frac{4}{5}=\frac{4}{5}$$

58. Combine the reduced real and imaginary terms.

    $$\frac{4\times i}{5}-\frac{2}{5}$$

59. Expand the brackets and replace i² by −1.

    $$\frac{4\times i}{5}-\frac{2}{5}=i\times y+x$$

60. Equal complex numbers have equal real parts and equal coefficients of i. Match these separately.

    $$(-\frac{2}{5}=x,\frac{4}{5}=y)$$

61. Use the equation containing only the first variable.

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

62. Subtract the constant term from both sides.

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

63. Simplify the subtraction.

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

64. Divide both sides by the coefficient.

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

65. Simplify the fraction.

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

66. Use the other original equation to find the second variable.

    $$\frac{4}{5}=y$$

67. Subtract the constant from both sides.

    $$\frac{4}{5}-\frac{4}{5}=y-\frac{4}{5}$$

68. Simplify the two sides.

    $$0=y-\frac{4}{5}$$

69. Divide both sides by the coefficient.

    $$y=\frac{y-\frac{4}{5}}{0}$$

70. Simplify to find the second value.

    $$y=\frac{4}{5}$$

71. Check both values in the original complex equation.

    $$\frac{4\times i}{5}-\frac{2}{5}=\frac{4\times i}{5}-\frac{2}{5}$$

Answer

The final answer is shown below.