The document discusses the discriminant of a quadratic equation and what it reveals about the nature of the roots. The discriminant, Δ, is calculated as b2 - 4ac. If Δ > 0, there are two distinct real roots. If Δ = 0, there are two equal real roots. If Δ < 0, there are no real roots. Several examples of finding the discriminant of equations and describing the roots are shown. The values of k that would result in equal or non-real roots for particular equations are also determined. Finally, the value of a for which the line y = ax is tangent to a given circle is found by setting the discriminant of the equation equal to 0.

1. The Discriminant

2. The Discriminant
  b 2  4ac

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

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

5. The Discriminant
  b 2  4ac
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
  b 2  4ac
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
  b 2  4ac
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
  b 2  4ac
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. (i ) Describe the roots of;

9. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0

10. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0
  52  4  3 9 
 83  0

11. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0
  52  4  3 9 
 83  0
 no real roots

12. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0 b ) 2x 2  6 x  3  0
  52  4  3 9 
 83  0
 no real roots

13. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0 b ) 2x 2  6 x  3  0
  52  4  3 9    62  4  2  3
 83  0  60  0
 no real roots

14. The Discriminant
  b 2  4ac
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. (i ) Describe the roots of;
a) 3x 2  5 x  9  0 b ) 2x 2  6 x  3  0
  52  4  3 9    62  4  2  3
 83  0  60  0
 no real roots  two different, real, irrational roots

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

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

17. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0

18. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0
36  4k  0
k 9

19. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0
36  4k  0
k 9
b) x 2  4 x  2k  0 have unreal roots

20. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0
36  4k  0
k 9
b) x 2  4 x  2k  0 have unreal roots
unreal roots occur when   0

21. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0
36  4k  0
k 9
b) x 2  4 x  2k  0 have unreal roots
unreal roots occur when   0
i.e.  4   4  2k   0
2

22. (ii) Find the values of k which makes;
a ) x 2  6 x  k  0 have equal roots
equal roots occur when   0
i.e. 62  4k  0
36  4k  0
k 9
b) x 2  4 x  2k  0 have unreal roots
unreal roots occur when   0
i.e.  4   4  2k   0
2
16  8k  0
k 2

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

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

25. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0

26. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0
4  16k 2  0
1
k 
2
4

27. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0
4  16k 2  0
1
k 
2
4
1 1
 k
2 2

28. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0
4  16k 2  0
1
k 
2
4
1 1
 k
2 2
(iii ) For what value of a is the line y  ax a tangent to
the circle x 2  y 2  20 x  10 y  100  0?

29. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0
4  16k 2  0
1
k 
2
4
1 1
 k
2 2
(iii ) For what value of a is the line y  ax a tangent to
the circle x 2  y 2  20 x  10 y  100  0?
x 2  a 2 x 2  20 x  10ax  100  0

30. c) kx 2  2 x  4k  0 have real roots
real roots occur when   0
i.e. 22  4  k  4k   0
4  16k 2  0
1
k 
2
4
1 1
 k
2 2
(iii ) For what value of a is the line y  ax a tangent to
the circle x 2  y 2  20 x  10 y  100  0?
x 2  a 2 x 2  20 x  10ax  100  0
a 2
 1 x 2  10  2  a  x  100  0

31. line is a tangent when   0

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

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

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

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

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

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