If α and β are the roots of the quadratic equation ax2 + bx + c = 0, then: (1) The sum of the roots is -b/a; (2) The product of the roots is c/a; (3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.

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