1. Complex Equations

2. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal

3. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1  b1i  a2  b2i

4. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1  b1i  a2  b2i
then
a1  a2
b1  b2

5. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1  b1i  a2  b2i
then
a1  a2
b1  b2
e.g .i  x  iy  2  3i 4  2i 

6. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1  b1i  a2  b2i
then
a1  a2
b1  b2
e.g .i  x  iy  2  3i 4  2i 
 8  4i  12i  6
 14  8i

7. Complex Equations
If two complex numbers are equal, then their real parts are equal and
their imaginary parts are equal
i.e. If a1  b1i  a2  b2i
then
a1  a2
b1  b2
e.g .i  x  iy  2  3i 4  2i 
 8  4i  12i  6
 14  8i
 x  14 , y  8

8. ii  z  2iw  4  3i
2 z  iw  3  4i

9. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i

10. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i
3z  2  5i
2 5
z  i
3 3

11. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i
3z  2  5i
2 5
z  i
3 3
2 5
  i  2iw  4  3i
3 3

12. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i
3z  2  5i
2 5
z  i
3 3
2 5
  i  2iw  4  3i
3 3 10 4
2iw   i
3 3

13. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i
3z  2  5i
2 5
z  i
3 3
2 5
  i  2iw  4  3i
3 3 10 4
2iw   i
3 3
10 4
w 
6i 6
2 5
  i
3 3

14. ii  z  2iw  4  3i z  2iw  4  3i

2 z  iw  3  4i 4 z  2iw  6  8i
3z  2  5i
2 5
z  i
3 3
2 5
  i  2iw  4  3i
3 3 10 4
2iw   i
3 3
10 4
w 
6i 6
2 5
  i
3 3
2 5 2 5
w  i , z  i
3 3 3 3

15. iii  Find the quadratic equation with roots 4  i and 4-i

16. iii  Find the quadratic equation with roots 4  i and 4-i
  8

17. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17

18. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0

19. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i

20. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2

21. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i

22. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5

23. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12

24. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
6
b
a

25. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a

26. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a
36
a  2 5
2
a

27. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a
36
a  2 5
2
a
a 4  5a 2  36  0

28. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a
36
a  2 5
2
a
a 4  5a 2  36  0
a 2  9a 2  4  0

29. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a
36
a  2 5
2
a
a 4  5a 2  36  0
a 2  9a 2  4  0
a 2  9 or a 2  4

30. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2 6
a2      5
6 b
 a
 a
36
a  2 5
2
a
a 4  5a 2  36  0
a 2  9a 2  4  0
a 2  9 or a 2  4
a  3
 b  2

31. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2
 6  5 6
a 
2
 b
 a a
36
a  2 5
2
a
a 4  5a 2  36  0
a 2  9a 2  4  0
a 2  9 or a 2  4
a  3 no real solutions
 b  2

32. iii  Find the quadratic equation with roots 4  i and 4-i
  8   17
 equation is x 2  8 x  17  0
iv  a  ib  5  12i
a  ib   5  12i
2
a 2  2abi  b 2  5  12i
 a2  b2  5 2ab  12
2
 6  5 6
a 
2
 b
 a a
36
a  2 5
2
a
a 4  5a 2  36  0
a 2  9a 2  4  0
a 2  9 or a 2  4
a  3 no real solutions
 a  3, b  2 or a  3, b  2
 b  2

33. If x  iy  a  ib

34. If x  iy  a  ib
then a 2  b 2  x
2ab  y

35. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i

36. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12

37. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16

38. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
8
b
a

39. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a

40. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0

41. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0

42. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0
a 2  4 or a 2  16

43. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0
a 2  4 or a 2  16
a  2
 b  4

44. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0
a 2  4 or a 2  16
a  2 no real solutions
 b  4

45. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0
a 2  4 or a 2  16
a  2 no real solutions
 b  4
 12  16i  2  4i 

46. If x  iy  a  ib
then a 2  b 2  x
2ab  y
e.g. Find  12  16i
a 2  b 2  12 2ab  16
64 8
a  2  12
2
b
a a
a 4  12a 2  64  0
a 2
 4 a 2  16   0
a 2  4 or a 2  16 Exercise 4A; 17 to 21, 22a, 23bc,
a  2 no real solutions 24, 25bd, 26ac, 27acd, 28ac, 29abd
 b  4
Exercise 4G; 1 aceg, 3 to 7
 12  16i  2  4i 