1. Solving Quadratics
e.g . x 2  3 x  7  0

2. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
2
 3   19

2
x

3. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
2
 3   19

2
 3  19i

2
 3  19i
 3  19i
x
or x 
2
2
x

4. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2
 3  19i
 3  19i
x
or x 
2
2
OR
by completing the square
x 2  3x  7  0

5. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2
 3  19i
 3  19i
x
or x 
2
2
OR
by completing the square
x 2  3x  7  0
2
3  19
x

  0
2
4

2
 x  3   19 i 2  0


2
4


6. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2
 3  19i
 3  19i
x
or x 
2
2
OR
by completing the square
x 2  3x  7  0
2
3  19
x

  0
2
4

2
 x  3   19 i 2  0


2
4

3
19 
3
19 

i  x  
i  0
x 
2
2 
2
2 


7. Solving Quadratics
e.g . x 2  3 x  7  0
 3  9  28
x
2
 3   19

2
 3  19i

2
 3  19i
 3  19i
x
or x 
2
2
OR
by completing the square
x 2  3x  7  0
2
3  19
x

  0
2
4

2
 x  3   19 i 2  0


2
4

3
19 
3
19 

i  x  
i  0
x 
2
2 
2
2 

3
19
3
19
x 
i or x   
i
2
2
2
2

8. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions

9. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions
 x  z   x  z   x2  ( z  z) x  z z

10. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions
 x  z   x  z   x2  ( z  z) x  z z
 x 2  2 Re( z ) x  z z

11. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions
 x  z   x  z   x2  ( z  z) x  z z
 x 2  2 Re( z ) x  z z
iii  Find the quadratic equation with roots 4  i and 4-i

12. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions
 x  z   x  z   x2  ( z  z) x  z z
 x 2  2 Re( z ) x  z z
iii  Find the quadratic equation with roots 4  i and 4-i
  8
  17

13. Creating Quadratics
All quadratics with real coefficients have
conjugate pairs as solutions
 x  z   x  z   x2  ( z  z) x  z z
 x 2  2 Re( z ) x  z z
iii  Find the quadratic equation with roots 4  i and 4-i
  8
  17
 equation is x 2  8 x  17  0

14. Finding Square Roots
a  ib  x  iy

15. Finding Square Roots
a  ib  x  iy
 a  ib   x  iy
2
a 2  2abi  b 2  x  iy

16. Finding Square Roots
a  ib  x  iy
If
x  iy  a  ib
 a  ib   x  iy
then a 2  b 2  x
a 2  2abi  b 2  x  iy
2ab  y
2

17. Finding Square Roots
a  ib  x  iy
If
x  iy  a  ib
 a  ib   x  iy
then a 2  b 2  x
a 2  2abi  b 2  x  iy
2ab  y
2
e.g. Find  12  16i

18. Finding Square Roots
a  ib  x  iy
If
x  iy  a  ib
 a  ib   x  iy
then a 2  b 2  x
a 2  2abi  b 2  x  iy
2ab  y
2
e.g. Find  12  16i
a 2  b 2  12
2ab  16
8
b
a

19. Finding Square Roots
a  ib  x  iy
If
x  iy  a  ib
 a  ib   x  iy
then a 2  b 2  x
a 2  2abi  b 2  x  iy
2ab  y
2
e.g. Find  12  16i
a 2  b 2  12
64
2
a  2  12
a
2ab  16
8
b
a

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

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

22. Finding Square Roots
a  ib  x  iy
If
x  iy  a  ib
 a  ib   x  iy
then a 2  b 2  x
a 2  2abi  b 2  x  iy
2ab  y
2
e.g. Find  12  16i
a 2  b 2  12
64
2
a  2  12
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
2ab  16
8
b
a

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

24. OR
a  ib  x  iy

25. OR
a  ib  x  iy
a 2  b2  x
2ab  y

26. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4

27. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4

28. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign

29. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2
e.g. Find  12  16i

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign

30. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2
e.g. Find  12  16i
a 2  b 2  12
a 2  b 2  20

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign

31. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2
e.g. Find  12  16i
a 2  b 2  12
a 2  b 2  20
2a 2
8
a2  4
a2

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign

32. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2
e.g. Find  12  16i
a 2  b 2  12
a 2  b 2  20
2a 2
8
a2  4
a  2 b  4

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign

33. OR
a  ib  x  iy
a b  x
2ab  y
2
x  y  a  b
2
2
If x  iy  a  ib
then a 2  b 2  x
a 2  b2  x2  y 2

2
2

2 2
 4a 2b 2
 a 4  2a 2 b 2  b 4  4a 2b 2
= a 4  2a 2 b 2  b 4
Note: y > 0, then a and b have same sign
y < 0, then a and b have different sign
e.g. Find  12  16i
a 2  b 2  12
a 2  b 2  20
8
2a 2
a2  4
a  2 b  4
 12  16i  2  4i 

34. (ii ) Find 5  4i

35. (ii ) Find 5  4i
a 2  b2  5
a 2  b 2  41

36. (ii ) Find 5  4i
a 2  b2  5
a 2  b 2  41
2a 2
 5  41
1
2
a  5  41
2

a

5  41
2
1
a
10  2 41
2

37. (ii ) Find 5  4i
a 2  b2  5
a 2  b 2  41
2a 2
 5  41
1
2
a  5  41
2

a

5  41
2
1
a
10  2 41
2



1
 b  5  41  5
2
1
 5  41
2
2

5  41
b
2
1
10  2 41
b
2

38. (ii ) Find 5  4i
a 2  b2  5
a 2  b 2  41
2a 2
 5  41
1
2
a  5  41
2

a

5  41
2
1
a
10  2 41
2
1
5  4i  
2
 10  2



1
 b  5  41  5
2
1
 5  41
2
2

5  41
b
2
1
10  2 41
b
2
41  i 10  2 41


39. (iii ) Solve z 2  (2  i ) z  (2  2i )  0

40. (iii ) Solve z 2  (2  i ) z  (2  2i )  0
z


2  i 
 2  i   4  2  2i 
2
2
  2  i   4  4i  1  8  8i
2
  2  i   5  12i
2

41. (iii ) Solve z 2  (2  i ) z  (2  2i )  0
z


2  i 
 2  i   4  2  2i 
2
2
  2  i   4  4i  1  8  8i
2
  2  i   5  12i
2
5  12i
a 2  b 2  5
a 2  b 2  13
2a 2
8
a2  4
a  2 b  3

42. (iii ) Solve z 2  (2  i ) z  (2  2i )  0
z



2  i 
 2  i   4  2  2i 
2
2
  2  i   4  4i  1  8  8i
2
  2  i   5  12i
2
  2  i    2  3i 
2
5  12i
a 2  b 2  5
a 2  b 2  13
2a 2
8
a2  4
a  2 b  3

43. (iii ) Solve z 2  (2  i ) z  (2  2i )  0
z



2  i 
 2  i   4  2  2i 
2
2
  2  i   4  4i  1  8  8i
2
  2  i   5  12i
2
  2  i    2  3i 
2
2i
4  4i

or
2
2
 i or  2  2i
5  12i
a 2  b 2  5
a 2  b 2  13
2a 2
8
a2  4
a  2 b  3

44. Cambridge: Exercise 1B; 1 to 4 ace etc, 5, 6, 7 cf,
9, 10, 16 ac