Obj. 3 Quadratic Equations (Presentation)

7/31/2019 Obj. 3 Quadratic Equations (Presentation)

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Obj. 3 Quadratic Equations

Unit 1 Functions and Relations


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Concepts and Objectives

Quadratic Equations

Simplify complex numbers Solve quadratic equations, finding all solutions

Cubic Equations

Solve the sum or difference of two cubes


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Complex Numbers

As you should recall, there is no real number solution to

x2 = 1so the number i has been defined so that

i2 = 1

which means that . Complex numbers arenumbers in the form a + bi, where a and b are real

numbers.

For any positive real number a,

= 1i

=a i a


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Complex Numbers

Example: Simplify (a) (b) (c)

(a)

(b)

(c)

16 48 ( )( ) 4 9

= =16 16 4i i

= =48 48 4 3i i

( )( ) = =4 9 36 6


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Quadratic Equations

A quadratic equation is an equation that can be written

in the form

where a, b, and c are real numbers, with a 0. This is

standard form.

A quadratic equation can be solved by factoring,

graphing, completing the square, or by using the

quadratic formula.

Graphing and factoring dont always work, butcompleting the square and the quadratic formula will

always provide the solution(s).

+ + =2

0ax bx c


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Factoring Quadratic Equations

Factoring works because of thezero-factor property:

Ifa and b are complex numbers with ab = 0, then a =0 or b = 0 or both.

To solve a quadratic equation by factoring:

Put the equation into standard form (=0). Find two numbers which multiply to ac and add to b.

Split the b term using these two numbers.

Find the GCF of each pair.

Use the distributive property to re-write the equation

into the factors.

Set each factor equal to zero and solve.


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Factoring Quadratic Equations

Example: Solve by factoring.

The solution set is

=2

2 15 0x x

= = = 2, 1, 15a b c 30

1

6 5 + =22 6 5 15 0x x x

( ) ( ) + =2 3 5 3 0x x x

( )( )+ =2 5 3 0x x

+ = =2 5 0 or 3 0x x

= 5

, 32x

5

,32


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Quadratic Formula

The solutions of the quadratic equation ,

where a 0, are

Example: Solve

+ + =2

0ax bx c

=

24

2

b b acx

a

= 2

2 4x x


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Quadratic Formula

Example: Solve

The solution set is

= 2

2 4x x

+ =22 4 0x x = = =2, 1, 4a b c

( ) ( ) ( )( )

( )

=

2

1 1 4 2 4

2 2x

= =

1 1 32 1 31

4 4

=

1 31

4

i

1 31

4 4i


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Cubic Equations

A cubic equation is an equation of degree 3. We will

mainly be working with cubic equations that are the sumor difference of two cubes:

a3 b3 = 0

Equations of this form factor as

To solve this, set each factor equal to zero and solve.

(Use the Quadratic Formula for the quadratic factor.)

( )( ) + =2 2 0a b a ab b


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Cubic Equations

Example: Solve 8x3 + 125 = 0 ( ) + =3 3

2 5 0x

( )( )+ + =22 5 4 10 25 0x x x

+ = + =2

2 5 0 or 4 10 25 0x x x

+ =

=

=

2 5 0

2 5

5

2

x

x

x

( ) ( )( )( )

=

2

10 10 4 4 252 4

x

= = =

10 300 10 10 3 5 5 3

8 8 4

i i

5 5 5 3,

2 4 4

iThe solution set is


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Homework & Practice Problems

Page 109: 20-40 (5s)

Page 119: 15-30 (5s), 45-55 (5s), 62, 64

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Obj. 3 Quadratic Equations (Presentation)