The document discusses complex numbers and equations. It states that for two complex numbers to be equal, their real parts and imaginary parts must be equal. It then provides an example of solving a complex equation by equating real and imaginary parts. Another example finds the quadratic equation with roots of 4+i and 4-i. A final example expresses a complex number as a quadratic equation.

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 