The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. It gives the following information: - The discriminant (Δ) tells us whether the roots are real or imaginary. - If Δ > 0, there are two different real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are no real roots. - If Δ is a perfect square, the roots are rational numbers. - Several examples of quadratic equations are worked through to demonstrate applying the discriminant. - Conditions are identified for quadratic equations to have equal, unreal or real roots based on the value of Δ.

1. The Discriminant

2. The Discriminant
2
4b ac  

3. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational

4. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)

5. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)

6. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)

7. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational

8. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i

9. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  

10. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  
  2
5 4 3 9
83 0
  
  

11. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  
  2
5 4 3 9
83 0
  
  
no real roots

12. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  
  2
5 4 3 9
83 0
  
  
no real roots
2
) 2 6 3 0b x x  

13. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  
  2
5 4 3 9
83 0
  
  
no real roots
2
) 2 6 3 0b x x  
  2
6 4 2 3
60 0
   
 

14. The Discriminant
2
4b ac  
The discriminant tells us whether the roots are rational or irrational
0  : two different real roots (cuts the x axis twice)
0  : two equal real roots (touches the x axis once)
0  : no real roots (never touches the x axis)
is a perfect square : roots are rational
e.g. ( ) Describe the roots of;i
2
) 3 5 9 0a x x  
  2
5 4 3 9
83 0
  
  
no real roots
2
) 2 6 3 0b x x  
  2
6 4 2 3
60 0
   
 
two different, real, irrational roots

15. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  

16. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 

17. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 

18. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 
36 4 0
9
k
k
 


19. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 
36 4 0
9
k
k
 

2
) 4 2 0 have unreal rootsb x x k  

20. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 
36 4 0
9
k
k
 

2
) 4 2 0 have unreal rootsb x x k  
unreal roots occur when 0 

21. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 
36 4 0
9
k
k
 

2
) 4 2 0 have unreal rootsb x x k  
unreal roots occur when 0 
   
2
. . 4 4 2 0i e k  

22. (ii) Find the values of k which makes;
2
) 6 0 have equal rootsa x x k  
equal roots occur when 0 
2
. . 6 4 0i e k 
36 4 0
9
k
k
 

2
) 4 2 0 have unreal rootsb x x k  
unreal roots occur when 0 
   
2
. . 4 4 2 0i e k  
16 8 0
2
k
k
 


23. 2
) 2 4 0 have real rootsc kx x k  

24. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 

25. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 

26. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 
2
2
4 16 0
1
4
k
k
 


27. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 
2
2
4 16 0
1
4
k
k
 

1 1
2 2
k  

28. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 
2
2
4 16 0
1
4
k
k
 

1 1
2 2
k  
2 2
( ) For what value of is the line a tangent to
the circle 20 10 100 0?
iii a y ax
x y x y

    

29. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 
2
2
4 16 0
1
4
k
k
 

1 1
2 2
k  
2 2
( ) For what value of is the line a tangent to
the circle 20 10 100 0?
iii a y ax
x y x y

    
2 2 2
20 10 100 0x a x x ax    

30. 2
) 2 4 0 have real rootsc kx x k  
real roots occur when 0 
  2
. . 2 4 4 0i e k k 
2
2
4 16 0
1
4
k
k
 

1 1
2 2
k  
2 2
( ) For what value of is the line a tangent to
the circle 20 10 100 0?
iii a y ax
x y x y

    
2 2 2
20 10 100 0x a x x ax    
   2 2
1 10 2 100 0a x a x    

31. line is a tangent when 0 

32. line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   

33. line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   
2 2
400 400 100 400 400 0a a a    

34. line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   
2 2
400 400 100 400 400 0a a a    
2
3 4 0a a 

35. line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   
2 2
400 400 100 400 400 0a a a    
2
3 4 0a a 
 3 4 0a a  

36. line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   
2 2
400 400 100 400 400 0a a a    
2
3 4 0a a 
 3 4 0a a  
4
0 or
3
a a  

37. Exercise 8F; 1ace, 2bdf, 3bg, 4ch, 5ad, 6, 7ac, 8be, 9ac,
11, 12b, 13, 14, 18, 21bd
line is a tangent when 0 
    
2 2
i.e. 100 2 4 1 100 0a a   
2 2
400 400 100 400 400 0a a a    
2
3 4 0a a 
 3 4 0a a  
4
0 or
3
a a  