Post on 25-May-2015
4-8 The Quadratic Formula 4-8 The Quadratic Formula and the Discriminantand the Discriminant
p.278-281p.278-281
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
The Quadratic Formula
Solve for x by completing the square.
222
2
2
2
22
0
a
b
a
c
a
bxa
bx
a
cxa
bx
cbxax
cbxax
The Quadratic Formula
Solve for x by completing the square.
222
2
2
2
22
0
a
b
a
c
a
bxa
bx
a
cxa
bx
cbxax
cbxax
2
22
2
2
2
22
4
4
2
4
4
4
a
acb
a
bx
a
acb
a
bxa
bx
The Quadratic Formula
Solve for x by completing the square.
2
22
2
2
2
22
4
4
2
4
4
4
a
acb
a
bx
a
acb
a
bxa
bx
a
acbbx
a
acb
a
bx
a
acb
a
bx
2
4
2
4
2
4
4
2
2
2
2
2
The Quadratic Formula
Solve for x by completing the square.
a
acbbx
2
42
How does it work
Equation:
1
5
3
0153 2
c
b
a
xx
a
acbbx
2
42
How does it work
Equation:
1
5
3
0153 2
c
b
a
xx
6
13
6
5
6
135
6
12255
32
13455 2
x
x
x
a
acbbx
2
42
The Discriminant
The number in the square root of the quadratic formula.
acb 42
12425
6145
0652
2
xxGiven
The Discriminant
The Discriminant can be negative, positive or zero
If the Discriminant is positive then 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
52
acb 42
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.
acb 42
Let’s put all of that information in a chart.
Value of DiscriminantType and
Number of RootsSample Graph
of Related Function
D > 0,D is a perfect square
2 real, rational roots
(ex: x= 2 and x= -4)
D > 0,D NOT a perfect square
2 real,Irrational roots
(x = √13 x= -√13)
D = 01 real, rational root
(double root)(ex: x = 5)
D < 02 complex roots
(complex conjugates)(x = 2 ± 3i )
acb 42
Describe the roots
Tell me the Discriminant and the type of roots 0962 xx
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
0962 xx
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
80, Two irrational roots
0962 xx
0532 xx
0482 xx
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
0962 xx
0532 xx
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
0962 xx
0532 xx
Describe the roots
Tell me the Discriminant and the type of roots
0, One rational root
-11, Two complex roots
0962 xx
0532 xx
0482 xx
Solve using the Quadratic formula
3382 xx
Solve using the Quadratic formula
12
331488
0338
338
2
2
2
x
xx
xx
Solve using the Quadratic formula
32
6
2
148
112
22
2
1482
148
2
1968
12
331488
0338
338
2
2
2
x
x
x
x
xx
xx
Solve using the Quadratic formula
0289342 xx
Solve using the Quadratic formula
12
289143434
028934
2
2
x
xx
Solve using the Quadratic formula
172
34
2
034
2
1156115634
12
289143434
028934
2
2
x
x
x
xx
Solve using the Quadratic formula
0262 xx
Solve using the Quadratic formula
732
72
2
6
2
286
2
8366
12
21466
026
2
2
x
x
x
xx
Solve using the Quadratic formula
12
131466
0136
613
2
2
2
x
xx
xx
Solve using the Quadratic formula
2
166
2
52366
12
131466
0136
613
2
2
2
x
x
xx
xx
Solve using the Quadratic formula
ix
ii
x
x
x
xx
xx
23
2
4
2
6
2
46
2
166
2
52366
12
131466
0136
613
2
2
2