The document discusses properties of the sum and product of roots of quadratic equations. It shows that: - The sum of the roots α and β is equal to -b/a - The product of the roots is equal to c/a - Examples are given of forming quadratic equations with given roots and calculating properties like the sum and product of roots for specific equations.

1. Sum and Product of Roots

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

23. 2 2
)c  

24. 2 2
)c    
2
2    

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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