ObjectiveSolving Quadratic Equations by the

Quadratic Formula

•Intro:Intro: We already know the We already know the standard form of a quadratic standard form of a quadratic equation is: equation is:

y = y = aaxx22 + + bbx + x + cc•The The constantsconstants are: are: a , b, ca , b, c•The The variablesvariables are: are: y, xy, x

•The The ROOTSROOTS (or (or solutionssolutions) of a ) of a polynomial are polynomial are its its x-interceptsx-intercepts

•Recall: The Recall: The x-x-interceptsintercepts occur where occur where y y = 0= 0..

Roots

•Example:Example: Find Find the roots: the roots: y = xy = x22 + x - 6+ x - 6

•Solution:Solution: Factoring: Factoring: y = (x y = (x + 3)(x - 2)+ 3)(x - 2)

•0 = (x + 3)(x - 2)0 = (x + 3)(x - 2)•The roots are: The roots are: •x = -3; x = 2x = -3; x = 2

Roots

•But what about But what about NASTYNASTY trinomials trinomials that don’t that don’t factor?factor?

•After centuries of After centuries of work, work, mathematicians mathematicians realized that as realized that as long as you know long as you know the coefficients, the coefficients, you can find the you can find the roots of the roots of the quadratic. Even if quadratic. Even if it doesn’t factor!it doesn’t factor!

y ax2 bx c, a 0

x b b2 4ac

2a

Example #1- continued

22 7 11 0x x

2

2

4

2

7 7 4(2)( 11)

2(

2, 7, 11

7 137 2 Reals - Irrational

4

2)

a b

b ac

a

c

b

Solve using the Quadratic Formula

Solve: y = 5x2 8x 3

x b b2 4ac

2a

a 5, b 8, c 3

x ( 8) ( 8)2 4(5)(3)

2(5)

x 8 64 60

10

x 8 4

10

x 8 2

10

x 8 2

10

x 8 2

10

10

101

x 8 2

10

6

10

3

5Roots

y 5(1)2 8(1) 3y 5 8 3y 0

y 5 35 2

8 35 3

y 5 925 24

5 3

y 4525 24

5 3

y 95 24

5 155

y 0

Plug in your Plug in your answers for answers for xx..If you’re right, If you’re right, you’ll get you’ll get y = y = 00..

Solve : y 2x2 7x 4

a 2, b 7, c 4

x b b2 4ac

2a

x (7) (7)2 4(2)( 4)

2(2)

x 7 49 32

4

x 7 81

4

x 7 9

4x

2

4

1

2

x 16

4 4

RememberRemember: All the terms must : All the terms must be on be on one sideone side BEFORE you use BEFORE you use the quadratic formula.the quadratic formula.

•Example:Example: Solve Solve 3m3m22 - 8 = 10m - 8 = 10m

•Solution:Solution: 3m3m2 2 - 10m - 8 = 0- 10m - 8 = 0

•a = 3, b = -10, c = -8a = 3, b = -10, c = -8

•Solve:Solve: 3x 3x22 = 7 - 2x = 7 - 2x•Solution:Solution: 3x 3x22 + 2x - 7 = 0 + 2x - 7 = 0•a = 3, b = 2, c = -7a = 3, b = 2, c = -7

x b b2 4ac

2a

x (2) (2)2 4(3)( 7)

2(3)

x 2 4 84

6

x 2 88

6

x 2 4 • 22

6

x 2 2 22

6x

1 22

3

WHY USE THE QUADRATIC FORMULA?

The quadratic formula allows you to solve ANY The quadratic formula allows you to solve ANY

quadratic equation, even if you cannot factor quadratic equation, even if you cannot factor

it.it.

An important piece of the quadratic formula is An important piece of the quadratic formula is

what’s under the radical:what’s under the radical:

bb22 – 4ac – 4ac

This piece is called the This piece is called the discriminantdiscriminant..

WHY IS THE DISCRIMINANT IMPORTANT?

The The discriminantdiscriminant tells you the tells you the numbernumber and and types of types of

answersanswers

(roots)(roots) you will get. The you will get. The discriminantdiscriminant can be +, –, or 0 can be +, –, or 0

which actually tells you a lot! Since the which actually tells you a lot! Since the discriminantdiscriminant is is

under a radical, think about what it means if you have a under a radical, think about what it means if you have a

positive or negative number or 0 under the radical.positive or negative number or 0 under the radical.

WHAT THE DISCRIMINANT TELLS YOU!

Value of the Discriminant Nature of the Solutions

Negative 2 imaginary solutions

Zero 1 Real Solution

Positive – perfect square 2 Reals- Rational

Positive – non-perfect square

2 Reals- Irrational

Example #1

22 7 11 0x x

Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula)

1.

a=2, b=7, c=-11

Discriminant = 2

2

4

(7) 4(2)( 11)

49

137

88

b ac

Discriminant =

Value of discriminant=137

Positive-NON perfect square

Nature of the Roots – 2 Reals - Irrational

Solving Quadratic Equations

by the Quadratic Formula

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,7

2.(6, 14)

3. 3,8

7 34.

2

5 475.

6

i

i