M3U3D5 Warm UpM3U3D5 Warm UpMultiply:

(-5 + 2i)(6 – 4i)

-30 +20i + 12i – 8i2

-30 + 32i – 8(-1)-30 + 32i + 8

-22 + 32i

F O I L

Remember the cycle of “i”

i0 = 1 i2 = -1i1 = i i3 = -i

Homework Check:

Homework Check:

Quadratic Formula, Discriminant, and Zeroes

OBJ: Solve quadratic equations with real coefficients that have

complex solutions.

THE QUADRATIC FORMULATHE QUADRATIC FORMULA

1. When you solve using completing the square on the general formula you get:

2. This is the quadratic formula!

3. Just identify a, b, and c then substitute into the

formula.

2 4

2

b b acx

a

2 0ax bx c

2 4

2x

cb b a

a

The Quadratic Formula Derived

02 cbxax

cbxax 2

a

cx

a

bx

a

a 2

a

cx

a

bx

2

a

b

a

b

22

1 2

22

42 a

b

a

b

a

c

a

b

a

bx

a

bx

2

2

2

22

44

a

a

a

c

a

b

a

bx

a

bx

4

4

44 2

2

2

22

2 4

2x

cb b a

a

The Quadratic Formula Derived

22

2

2

22

4

4

44 a

ac

a

b

a

bx

a

bx

2

2

2

22

4

4

4 a

acb

a

bx

a

bx

2

2

2

22

4

4

4 a

acb

a

bx

a

bx

2

22

4

4

2 a

acb

a

bx

2

2

4

4

2 a

acb

a

bx

a

acb

a

bx

2

4

2

2

a

acb

a

bx

2

4

2

2

a

acbbx

2

42

The quadratic formula is used to solve any quadratic equation.

2 4

2x

cb b a

a

The quadratic formula is:

What are a, b, and c?

Standard form of a quadratic equation is: 2 0x xba c

2 4 8 0x x

a 1 c b4 8

23 5 6 0x x

a 3 c b 5

22 0x x

a 2 c b1 0

2 10x a 1 c b0 106

2 10 0x

The Quadratic Formula

2 4

2x

cb b a

a

2 0x xba c

2 3 2 0x x

2x 1x

1x 2x 0

1 0x 2 0x

The Quadratic Formula

Solve the problem by factoring.

2 4

2x

cb b a

a

2 0x xba c

2 3 2 0x x a 1 c b 3 2

23 3 1 24

12x

3 9 8

2x

3 1

2x

3 1

2x

3 1

2x

3 1

2x

4

2x

2x

2

2x

1x 3 1

2x

Solve the same problem using The Quadratic Formula

2 4

2x

cb b a

a

2 0x xba c

22 5 0x x

a 2 c b 1 5

24

22

1 521x

1 1 40

4x

1 41

4x

1. You try.

2 4

2x

cb b a

a

2. Another example.

44 2 xx

044 2 xx

42

44411 2 x

8

6411 x

8

631 x

8

631 ix

8

791

ix

8

731 ix

ix

8

73

8

1

WHY USE THE WHY USE THE

QUADRATIC FORMULA?QUADRATIC FORMULA?

• The quadratic formula allows you to solve ANY

quadratic equation, even if you cannot factor it.

• An important piece of the quadratic formula is

what’s under the radical:

b2 – 4ac

• This piece is called the discriminant.

WHY IS THE DISCRIMINANT WHY IS THE DISCRIMINANT

IMPORTANT?IMPORTANT?

The discriminant tells you the number and types of answers

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

which actually tells you a lot! Since the discriminant is

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

positive or negative number or 0 under the radical.

WHAT THE DISCRIMINANT WHAT THE DISCRIMINANT

TELLS YOU!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 #1Example #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

Example #1- continuedExample #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

Solving Quadratic Equations Solving Quadratic Equations

by the Quadratic Formulaby 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

2 4

2x

cb b a

a

The Quadratic Formula and the Discriminant

REMEMBER, the discriminant is the radicand portion of the quadratic formula (b2 – 4ac).

It is used to discriminate among the possible number and type of solutions a quadratic equation will have.

b2 – 4acName and Type of

SolutionPositive

Zero

Negative

Two real solutions

One real solutions

Two complex, non-real solutions

2 4

2x

cb b a

a

The Quadratic Formula and the Discriminate

2143 2

89

b2 – 4acName and Type of

SolutionPositive

Zero

Negative

Two real solutions

One real solutions

Two complex, non-real solutions

2 3 2 0x x a 1 c b 3 2

1

Positive

Two real solutions

2x 1x

2 4

2x

cb b a

a

The Quadratic Formula and the Discriminate

4441 2

641

b2 – 4acName and Type of

SolutionPositive

Zero

Negative

Two real solutions

One real solutions

Two complex, non-real solutions

a c b

63

Negative

Two complex, non-real solutions

044 2 xx

4 1 4

ix8

73

8

1

2(25)

4(25)(36)- 60)(- 60)(- - x

2

25x 25x 22 - 60x + 36 = 0 - 60x + 36 = 0

18 x 30- x 225 2 f(x)

Find the zeros using the Find the zeros using the

Quadratic FormulaQuadratic Formula

2(25)

4(25)(36)- 60)(- 60)(- - x

2

Exact Solution:Exact Solution:

56

50

0 60 x

2(25)

4(25)(36)- 60)(- 60)(- - x

2

Calculator Calculator Solution:Solution: x = 1.2x = 1.2

Check Check Intercepts!Intercepts!

Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:

f(x) = 3x f(x) = 3x 22 + 2 - 4x + 2 - 4x

3x 3x 22 - 4x + 2 = 0 - 4x + 2 = 0

2(3)

4(3)(2)- 4)(- 4)(- - x

2

Find the zeros using the Find the zeros using the

Quadratic Formula…You try!Quadratic Formula…You try!

2(3)

4(3)(2)- 4)(- 4)(- - x

2

6

8- 4 x

No real solution. No real solution.

Check Intercepts.Check Intercepts.

2 4

2x

cb b a

a

A real world application of The Quadratic Formula:

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

2 4

2x

cb b a

a

The Quadratic Formula

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

(x + 2)2 + x2 = 202

x2 + 4x + 4 + x2 = 400

2x2 + 4x + 4 = 400

2x2 + 4x – 396 = 0

2(x2 + 2x – 198) = 0

2 4

2x

cb b a

a

The Quadratic Formula

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

2(x2 + 2x – 198) = 0

12

1981422 2 x

2

79242 x

2

7962 x

2 4

2x

cb b a

a

The Quadratic Formula

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

2

7962x

2

2.282

2

2.282 x

2

2.282 x

2

2.26x

1.13x

2

2.30x

1.15xft

2 4

2x

cb b a

a

The Quadratic Formula

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

1.13x

ft2.28

ft

21.131.13

28 – 20 = 8 ft

ClassworkClasswork U3D5 Packet Pages 1 & 2

HomeworkHomeworkU3D5 Packet Pages 3 & 4 odds