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
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
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
a2
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