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