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5-6 The Quadratic Formula 5-6 The Quadratic Formula and the Discriminantand the Discriminant
p.292-297p.292-297
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
−±−=
−±−=
−±=+
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
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
( ) ( ) ( ) ( )( )
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
( ) ( ) ( ) ( )
( )
172
34
2
034
2
1156115634
12
289143434
028934
2
2
==±=
−±=
−−±−−=
=+−
x
x
x
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
( ) ( ) ( )( )( )
2
166
2
52366
12
131466
0136
613
2
2
2
−±=−±=
−−±−−=
=+−=+
x
x
xx
xx