1. Sum and Product of Roots

2. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;

3. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   

4. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   
ax 2  bx  c  a  x 2   x   x   

5. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   
ax 2  bx  c  a  x 2   x   x   
b c
x 2  x   x 2      x  
a a

6. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   
ax 2  bx  c  a  x 2   x   x   
b c
x 2  x   x 2      x  
a a
Thus

7. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   
ax 2  bx  c  a  x 2   x   x   
b c
x 2  x   x 2      x  
a a
Thus
b
   (sum of roots)
a

8. Sum and Product of Roots
If  and  are the roots of ax 2  bx  c  0, then;
ax 2  bx  c  a  x    x   
ax 2  bx  c  a  x 2   x   x   
b c
x 2  x   x 2      x  
a a
Thus
b
   (sum of roots)
a
c
  (product of roots)
a

9. e.g. (i) Form a quadratic equation whose roots are;

10. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3

11. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3
    1

12. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3
    1
  6

13. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3
    1
  6
x2  x  6  0

14. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1
  6
x2  x  6  0

15. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6
x2  x  6  0

16. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1

17. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0

18. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0
(ii ) If  and  are the roots of 2x 2  3 x  1  0, find;

19. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0
(ii ) If  and  are the roots of 2x 2  3 x  1  0, find;
a)   

20. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0
(ii ) If  and  are the roots of 2x 2  3 x  1  0, find;
b
a)    
a
3

2

21. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0
(ii ) If  and  are the roots of 2x 2  3 x  1  0, find;
b b) 
a)    
a
3

2

22. e.g. (i) Form a quadratic equation whose roots are;
a ) 2 and  3 b) 2  5 and 2  5
    1   4
  6   4  5
x2  x  6  0  1
x2  4x 1  0
(ii ) If  and  are the roots of 2x 2  3 x  1  0, find;
b b)  
c
a)    
a a
3 1
 
2 2

23. c)  2   2

24. c)  2   2       2
2

25. c)  2   2       2
2
1
2
    2 
3
   
 2  2

26. c)  2   2       2
2
1
2
    2 
3
   
 2  2
9
 1
4
13

4

27. 1 1
c)          2 
2 2 2
d)
 
1
2
    2 
3
   
 2  2
9
 1
4
13

4

28. 1 1  
c)          2 d)  
2 2 2
  
1
2
    2 
3
   
 2  2
9
 1
4
13

4

29. 1 1  
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13

4

30. 1 1  
c)          2 d)  
2 2 2
  
 1 
2
3 3
    2 
2  2  2
9 1
 1 2
4
13  3

4

31. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0

32. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2

33. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6

34. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6
3  6
  2

35. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6   2   m
3  6
  2

36. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6   2   m
3  6 2 2  m
  2

37. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6   2   m
3  6 2 2  m
2  2   m
2
  2

38. 1  
1
c)          2 d)  
2 2 2
  
1
2
    2 
3 3
   
 2  2  2
9 1
 1 2
4
13  3

4
(iii ) Find the value of m if one root is double the other in x 2  6 x  m  0
Let the roots be  and 2
  2  6   2   m
3  6 2 2  m
2  2   m
2
  2
m8

39. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.

40. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.
1
Let the roots be  and


41. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.
1
Let the roots be  and

1 1
   
   2m  1

42. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.
1
Let the roots be  and

1 1
   
   2m  1
1
1
2m  1

43. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.
1
Let the roots be  and

1 1
   
   2m  1
1
1
2m  1
2m  1  1
2m  2
m 1

44. (iv) Find the values of m in  2m  1 x 2  1  m  x  1  0, if one root is the
reciprocal of the other.
1
Let the roots be  and

1 1
   
   2m  1
1
1
2m  1
2m  1  1
2m  2
m 1
Exercise 8H; 3beh, 4bdf, 5, 6, 7abc i, 8, 10, 12,
13cd, 15, 18, 20, 21ac