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

 3  9  28
2
 3   19

2

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

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

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

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

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

 x  z   x  z   x2  ( ...
Finding Square Roots
a  ib  x  iy
Finding Square Roots
a  ib  x  iy

 a  ib   x  iy
2

a 2  2abi  b 2  x  iy
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
OR

a  ib  x  iy
OR

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


...
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


...
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...
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...
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...
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...
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

...
(ii ) Find 5  4i
(ii ) Find 5  4i

a 2  b2  5
a 2  b 2  41
(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
(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




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



2  i 

 2  i   4  2  2i 
2

2
  2  i   4  4i  ...
(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...
Cambridge: Exercise 1B; 1 to 4 ace etc, 5, 6, 7 cf,
9, 10, 16 ac
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X2 t01 02 solving quadratics (2013)

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Transcript of "X2 t01 02 solving quadratics (2013)"

  1. 1. Solving Quadratics e.g . x 2  3 x  7  0
  2. 2. Solving Quadratics e.g . x 2  3 x  7  0  3  9  28 2  3   19  2 x
  3. 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. 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. 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. 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. 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. 8. Creating Quadratics All quadratics with real coefficients have conjugate pairs as solutions
  9. 9. Creating Quadratics All quadratics with real coefficients have conjugate pairs as solutions  x  z   x  z   x2  ( z  z) x  z z
  10. 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. 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. 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. 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. 14. Finding Square Roots a  ib  x  iy
  15. 15. Finding Square Roots a  ib  x  iy  a  ib   x  iy 2 a 2  2abi  b 2  x  iy
  16. 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. 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. 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. 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. 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. 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. 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. 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. 24. OR a  ib  x  iy
  25. 25. OR a  ib  x  iy a 2  b2  x 2ab  y
  26. 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. 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. 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. 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. 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. 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. 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. 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. 34. (ii ) Find 5  4i
  35. 35. (ii ) Find 5  4i a 2  b2  5 a 2  b 2  41
  36. 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. 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. 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. 39. (iii ) Solve z 2  (2  i ) z  (2  2i )  0
  40. 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. 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. 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. 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. 44. Cambridge: Exercise 1B; 1 to 4 ace etc, 5, 6, 7 cf, 9, 10, 16 ac
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