The Quadratic FormulaThe Quadratic Formula can be used to solve any quadratic equation that is in the form ax2 + bx + c = 0

I’ll have to write that 138 times before I’ll be able to remember it.

2 42

xcb b a

a

Let’s take a look at what this formula looks like.

Solving Equations with the Quadratic Formula

2 083 2x x

Find the roots of the following equation.

Let’s solve by factoring first.

4) 07( )(x x 7 0x 4 0x

7x 4x

7 4{ , }x

Now let’s use the quadratic formula.

2 42

xcb b a

a

3, , 1 28ca b

2( ) ( ) 4( )( )2( )

x

3 3 1 -281

3 9 1122

x

3 121

2

3 112

x

3 112

x 3 11

2x

82

x 4142

x

7

7 4{ , }x

Hey, it’s Sam Ting.

Solving Unfactorable Equations

Find the roots of the following equation.

2 12 03x x

I can’t factor that!

2 42

xcb b a

a

3 12, , ba c

2( ) ( ) 4( )( )2( )

x

-3 -3 2 -1

2

3 9 84

x 3 17

4

3 174

x

Find the roots of the same equation to the nearest hundredth.

3 174

x 3 4.123

4

3 4.1234

x

3 4.123

4x

7.1234

x

1.781x

1.1234

x

0.281x

0.28{ , 78}1.x

Finding Imaginary RootsFind the roots of the following equation in simplest a + bi form.

2 5 02x x 2 51, , ba c

2 42

xcb b a

a

2( ) ( ) 4( )( )2( )

x

-2 -2 1 5

1

2 4 202

x 2 16

2

2 42

xi

1 2x i

2 9 04x x 4 1, 9,ba c

2( ) ( ) 4( )( )2( )

x

4 4 1 9

1

4 16 362

x 4 20

2

4 2 52

ix

2 5x i

I think I understand.

Irrational and Imaginary Roots always occur in conjugate pairs.

One More Example2 9 06x x

2 42

xcb b a

a

6 91, , ba c

2( ) ( ) 4( )( )2( )

x

-6 -6 1 91

6 36 362

x

6 0

2

6 02

x

6 02

x 6 0

2x

62

x

3x

62

x

3x

Hey, it’s Sam Ting.

Using the DiscriminantThe Discriminant is the expression under the radical sign in the quadratic formula.

Let’s take another look at that formula.

2 42

xcb b a

a

2 4b ca

2 4b ca The Discriminant tells the nature of the roots.

The Nature of the Roots can be

Real, Rational, and Unequal

Real, Irrational, and Unequal

Real, Rational, and Equal

Imaginary

RR

RI

RR

I mag

I Think I’d like to see some examples.

Discriminant Examples2 083 2x x

2

2

4

( )

2( )3 1 84( )

acb

7 4{ , }x

2 1 14 2cab

2 12 03x x 2

2

4

(

3) 4(2 )1)(

ab c

2 174acb 0.28{ , 78}1.x

2 9 06x x 2

2

4

(

6 4 ) )91) ( (

b ca

2 4 0b ac

3{ }3, x

2 5 02x x 2

2

4

(

2 4 ) )51) ( (

b ca

2 164 cb a

{ , 2 }1 2 1x i i

Perfect Square

RR

NonPerfect Square

RI

Zero

RR

Negative

I maginary

Overview of the Discriminant

Perfect Square

RR Intersects x-axis at 2

Distinct Points

Non-Perfect Square

RI Intersects x-axis at 2

Non-Distinct Points

Zero

RR Intersects x-axis at 1

Distinct Point

NegativeI maginary

Does Not Intersect x-

axis

2 6 0x x

22 3 1 0x x

2 6 9 0x x

2 2 2 0x x

2 254acb

2 174acb

2 4 0b ac

2 4 4ab c

Sum and Product of the Roots

Equation & Roots Sum Product

2 083 2x x

7 4{ , }x ( ) (7 ) 34 7 4( )( ) 28

2 9 06x x 3{ }3, x

( ) ( 63 3) (3)(3) 9

2 12 03x x

3 174

x

3 17 3 174 4

64

32

3 17 3 174 4

8 1216

2 5 02x x

{ , 2 }1 2 1x i i ( ) (1 2 1 )2 2i i ( ) (1 2 1 )2 5i i

Do you need to solve the equation to find the sum and

product?

Formulas to Find the Sum and Product of the Roots

Sum of the Roots =ab

Product of the Roots =ca

Equation Sum Product

2 083 2x x 3(3)1

8281

2

2 9 06x x 1

(6

6) 91

9

2 12 03x x ( )

23 3

2

21

21

2 5 02x x 1

(2

2) 51

5

Notice that they are all the same answers that we

got before.

Problems Using the Sum and Product of the Roots

For the following quadratic equation, by what amount does the product of the roots exceed the sum of the roots?2 11 072x x

Sum of the Roots =( )

27 7

2 Product of the Roots =

112

112

Sab P

ac

11 7 42 2 2

2 The product of the roots exceeds the sum of the roots by 2.

If one root of the following quadratic equation is –1, find(a) The other root and (b) the value of k.

22 6 0x x k

Sab(a) 6)

3(2

S 1 3x 4x The other root is 4

(b) The two roots are –1 and 4. The sum is 3 and the product is –4.

Pac 4

2k 8k That was easy

Forming an Equation from the Roots

Form an equation whose roots are x = 2 and x = –5.

The factors are: ( 2)x and( 5)x 2( )( )2 105 3x x x x 2 is the quadratic 3 equatio1 n.0 0 x x

Form a quadratic equation that has 5 + 2i as one of its roots.

Since imaginary roots occur in conjugate pairs, the roots are:

and 5 2 5 2i i

5 2 5( ) ) 0( 2 1i i and 5 2 5( ) ) 9( 2 2i i

Sab Assume a =

1 10

1b 10b

Pac Assume a =

1 2

19

c 29c

2

is the

quadrati

29

c

equation.

0

10x x

Using Sum & Product of the Roots to Form an

EquationWrite a quadratic equation that has the given roots.

34 6

,5

Sum =

3 54 6

9 10

12 12

211

Product =

3 54 6

2415

242

41

,5

2S P

Sab P

ac

2 14 5 022 x x

23

,12

Sum =

2 13 2

4 36 6

67

Product =

2 13 2

6

2 2 26 07x x