Mathematics 10 — Exercise 1.2 · Q4 (iv)

Full question

For z₁ = 2 + 5i, z₂ = 1 − 3i, z₃ = 2 + i, verify the stated property by computing both sides.

Step-by-step solution

1. Compute the left side first.

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

   $$5\times i\times \left(1-3\times i\right)+2\times \left(1-3\times i\right)$$

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

   $$2\times 1+2\times \left(-1\right)\times 3\times i+5\times 1\times i+5\times \left(-3\right)\times ii$$

4. Real × real: multiply the numerical coefficients.

   $$2\times 1=2$$

5. Real × imaginary: multiply the numerical coefficients.

   $$2\times \left(-1\right)\times 3\times i=-6\times i$$

6. Imaginary × real: multiply the numerical coefficients.

   $$5\times 1\times i=5\times i$$

7. Imaginary × imaginary: multiply the numerical coefficients.

   $$5\times \left(-3\right)\times ii=-15\times i^2$$

8. Substitute these four products into the expansion.

   $$2-6\times i+5\times i-15\times i^2$$

9. In the last term replace i² by −1.

   $$-15\times i^2=\left(-15\right)\times \left(-1\right)$$

10. Multiply its coefficient by −1.

    $$\left(-15\right)\times \left(-1\right)=15$$

11. Rewrite the expansion after this sign calculation.

    $$2-6\times i+5\times i+15$$

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

    $$2+15+i\times \left(-6+5\right)$$

13. Calculate the sum of the real terms.

    $$2+15=17$$

14. Calculate the sum of the imaginary coefficients.

    $$-6+5=-1$$

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

    $$17-i$$

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

    $$-i\times \left(i+2\right)+17\times \left(i+2\right)$$

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

    $$17\times 2+17\times 1\times i-2\times i-1\times ii$$

18. Real × real: multiply the numerical coefficients.

    $$17\times 2=34$$

19. Real × imaginary: multiply the numerical coefficients.

    $$17\times 1\times i=17\times i$$

20. Imaginary × real: multiply the numerical coefficients.

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

21. Imaginary × imaginary: multiply the numerical coefficients.

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

22. Substitute these four products into the expansion.

    $$34+17\times i-2\times i-i^2$$

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

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

24. Multiply its coefficient by −1.

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

25. Rewrite the expansion after this sign calculation.

    $$34+17\times i-2\times i+1$$

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

    $$34+1+i\times \left(17-2\right)$$

27. Calculate the sum of the real terms.

    $$34+1=35$$

28. Calculate the sum of the imaginary coefficients.

    $$17-2=15$$

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

    $$15\times i+35$$

30. Now independently compute the right side.

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

    $$i+2-3\times i\times \left(i+2\right)$$

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

    $$1\times 2+1\times i-3\times 2\times i-3\times 1\times ii$$

33. Real × real: multiply the numerical coefficients.

    $$1\times 2=2$$

34. Real × imaginary: multiply the numerical coefficients.

    $$1\times i=i$$

35. Imaginary × real: multiply the numerical coefficients.

    $$-3\times 2\times i=-6\times i$$

36. Imaginary × imaginary: multiply the numerical coefficients.

    $$-3\times 1\times ii=-3\times i^2$$

37. Substitute these four products into the expansion.

    $$2+i-6\times i-3\times i^2$$

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

    $$-3\times i^2=\left(-3\right)\times \left(-1\right)$$

39. Multiply its coefficient by −1.

    $$\left(-3\right)\times \left(-1\right)=3$$

40. Rewrite the expansion after this sign calculation.

    $$2+i-6\times i+3$$

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

    $$2+3+i\times \left(1-6\right)$$

42. Calculate the sum of the real terms.

    $$2+3=5$$

43. Calculate the sum of the imaginary coefficients.

    $$1-6=-5$$

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

    $$5-5\times i$$

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

    $$5\times i\times \left(5-5\times i\right)+2\times \left(5-5\times i\right)$$

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

    $$2\times 5+2\times \left(-1\right)\times 5\times i+5\times 5\times i+5\times \left(-5\right)\times ii$$

47. Real × real: multiply the numerical coefficients.

    $$2\times 5=10$$

48. Real × imaginary: multiply the numerical coefficients.

    $$2\times \left(-1\right)\times 5\times i=-10\times i$$

49. Imaginary × real: multiply the numerical coefficients.

    $$5\times 5\times i=25\times i$$

50. Imaginary × imaginary: multiply the numerical coefficients.

    $$5\times \left(-5\right)\times ii=-25\times i^2$$

51. Substitute these four products into the expansion.

    $$10-10\times i+25\times i-25\times i^2$$

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

    $$-25\times i^2=\left(-25\right)\times \left(-1\right)$$

53. Multiply its coefficient by −1.

    $$\left(-25\right)\times \left(-1\right)=25$$

54. Rewrite the expansion after this sign calculation.

    $$10-10\times i+25\times i+25$$

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

    $$10+25+i\times \left(-10+25\right)$$

56. Calculate the sum of the real terms.

    $$10+25=35$$

57. Calculate the sum of the imaginary coefficients.

    $$-10+25=15$$

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

    $$15\times i+35$$

59. The results are identical, proving the equality.

    $$15\times i+35=15\times i+35$$

Answer

The final answer is shown below.