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.
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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
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