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

−±−=

−±−=

−±=+

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

±=

±=±=

−±=−±=

−−±−−=

=+−=+