educational

1. 6.5 The Quadratic Formula
and the Discriminant

2. We have a number of different way
of finding the roots if a quadratic
equations
#1. Making a table
#2. Factoring
#3. Completing the Square
Now a new way that comes from completing
the square.
The Quadratic Formula

3. The Quadratic Formula
Solve for x by completing the square.
a
ac
b
b
x
2
4
2





4. How does it work
Equation:
1
5
3
0
1
5
3 2






c
b
a
x
x
a
ac
b
b
x
2
4
2





5. How does it work
Equation:
1
5
3
0
1
5
3 2






c
b
a
x
x       
 
6
13
6
5
6
13
5
6
12
25
5
3
2
1
3
4
5
5
2














x
x
x
a
ac
b
b
x
2
4
2





6. The Discriminant
The number in the square root of the
quadratic formula.
ac
b 4
2

    
1
24
25
6
1
4
5
0
6
5
2
2






 x
x
Given

7. The Discriminant
The Discriminant can be negative, positive or zero
If the Discriminant is positive,
there are 2 real answers.
If the square root is not a prefect square
( for example ),
then there will be 2 irrational roots
( for example ).
25
5
2 

8. The Discriminant
The Discriminant can be negative, positive or zero
If the Discriminant is positive,
there are 2 real answers.
If the Discriminant is zero,
there is 1 real answer.
If the Discriminant is negative,
there are 2 complex answers.
complex answer have i.

9. Solve using the Quadratic formula
33
8
2

 x
x

10. Solve using the Quadratic formula
      
 
1
2
33
1
4
8
8
0
33
8
33
8
2
2
2












x
x
x
x
x

11. Solve using the Quadratic formula
      
 
3
2
6
2
14
8
11
2
22
2
14
8
2
14
8
2
196
8
1
2
33
1
4
8
8
0
33
8
33
8
2
2
2


























x
x
x
x
x
x
x
x

12. Solve using the Quadratic formula
0
289
34
2


 x
x

13. Solve using the Quadratic formula
      
 
1
2
289
1
4
34
34
0
289
34
2
2









x
x
x

14. Solve using the Quadratic formula
      
 
17
2
34
2
0
34
2
1156
1156
34
1
2
289
1
4
34
34
0
289
34
2
2
















x
x
x
x
x

15. Solve using the Quadratic formula
0
2
6
2


 x
x

16. Solve using the Quadratic formula
      
 
7
3
2
7
2
2
6
2
28
6
2
8
36
6
1
2
2
1
4
6
6
0
2
6
2
2


















x
x
x
x
x