Solving Quadratic Equations by FACTORING What are Quadratic Equations? A quadratic equation is an equation which: y = x 2 y = x 2 + 2 y = x 2 + x – 4

SolvingQuadratic Equations

by

FACTORING

What are Quadratic Equations?A quadratic equation is an equation

which:y = x2

y = x2 + 2y = x2 + x – 4y = x2 + 2x – 3

Contains a x2 termContains a x2 term

All of these equations contain a x2 term therefore they are called:

Quadratic Equations

Which of the following are

Quadratic Equations?

It Contains a x2 term


y = x + 3

y = x2

y = 2x – 4

y = x2 + 2x – 3

....WHY?

….WHY? It Contains a x2 term


Solving Quadratic Equations

BY FACTORING

Remember:Quadratic

Equations Contain a x2 term

Remember:Quadratic

Equations Contain a x2 term

There are several methods of solving QUADRATICS

but one methods that you must know is called

FACTORING

“Factors” are the numbers you multiply to get another number

The (+) factors of 6 are: 1 x 6 and 2 x 3

-1 x -6 and -2 x -3 The (-) factors of 6 are:


BY FACTORING

BIG IDEA NUMBER ONEIf A(B) = 0 what can we say about either

A or B?Either A or B must equal ZERO!!!

A = 0 or B = 0


BY FACTORING

BIG IDEA NUMBER ONE

(x + 3) (x – 3) = 0

So if…

THEN EITHER

(x + 3) = 0 or (x – 3) = 0

x = -3 or x = 3

So….


BY FACTORING

BIG IDEA NUMBER ONE

TO SOLVE A QUADRATIC EQUATION BY FACTORING

• MAKE THE EQUATION EQUAL TO ZERO

• FACTOR THE EQUATION

• SET THE FACTORS EQUAL TO ZERO AND SOLVE

How to solve Quadratic Equations by FACTORING

Example 1 x2 + x + = 0

Write down all the factor pairs of ___.

1 x 12 =12 -1 x -12 = 122 x 6 = 12 -2 x -6 = 123 x 4 = 12 -3 x -4 = 12

From this list, choose the pair that adds up to ___

3 + 4 = 7

Put these numbers into brackets

0 = (x + )(x + )

1

2

3

(x )(x ) = 0What goes with

the x?

(x )(x ) = 0What goes with

the x?Positive Negative

77 1212

0 = (x + 3)(x + 4)0 = (x + 3)(x + 4)

x = – 3 and – 4

(x + 3) (x + 4) x(x + 4) + 3(x + 4)

x(x) + x(4) + 3(x) + 3(4)

x2 + 4x + 3x + 12

x2 + 7x + 12

PROOF:

x2 + 7x + 12 = (x + 3) (x + 4)

Factor:

Factor:

Combine like terms:

How to solve Quadratic Equations by FACTORING

Example 2 x2 x + = 0Write down all

the factor pairs of ___ .

1 x 6 = 6 -1 x -6 = 62 x 3 = 6 -2 x -3 = 6

From this list, choose the pair that adds up to ___ .

-2 + -3 = -5

Put these numbers into brackets

(x - 2)(x - 3) = O

x = 2 and 3

1

2

3

Positive Negative

- 5- 5 66

Solve by factoring: x2 + x - 6 = 0

1 x -6 = -6 2 x -3 = -6 3 x -2 = -6 6 x -1 = -6

Write down all the factor pairs of – 6

From this list, choose the pair that adds up to 1

(3) + (-2) = 1

3 – 2 = 1Put these numbers into brackets

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

x = – 3 and 2

1

2

3

Copy and fill in the missing values

when you factor

x2 + 8x + 12 = 0

Find all the factor pairs of

_____

From these choose the pair that add up to _____

Put these values into the brackets (x _ )(x _ ) = 0

x = -2 x = -6

USE WORKSHEET #1

x2 + 3x + 2 = 0Find all the factor pairs of _____


Put these values into the brackets

(x _)(x _) = 0

x2 + x – 12 = 0Find all the factor pairs of _____



(x + _)(x + _) = 0

x2 – 12x – 20 = 0Find all the factor pairs of _____



(x + _)(x + _) = 0

.

12

8

+2

+6

2

3

+ 2

+ 1

1 x 2 = 2 -1 x -2 = 2

1 + 2= 3

1 x 12 =12 -1 x -12 = 122 x 6 = 12 -2 x -6 = 123 x 4 = 12 -3 x -4 = 12 2 + 6 = 8

PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS

POWERPOINT

WORKSHEET # 1

WORK TOGETHER TO FACTOR THE NEXT QUADRATIC

1 x2 + 5x + 6 = 0

2 x2 - x – 6 = 03 x2 + 8x + 12 =

04 x2 + x – 12 = 05 x2 - 8x + 15 =

06 x2 + 3x – 28 =

07 x2 - 3x – 18 =

08 x2 - 10x – 24 =

09 x2 + 8x + 16 =

010 x2 - 6x – 40 =

0

PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS

POWERPOINT

WORKSHEET # 2

(x + 3)(x + 2)(x – 3)(x + 2)

(x + 2)(x + 6)

(x – 3)(x + 4)

(x – 3)(x – 5)

(x + 7)(x – 4)(x – 6)(x + 3)

(x - 12)(x + 2)(x + 4)(x + 4)(x - 10)(x + 4)

1 x2 + 5x + 6 = 0

(x + 3)(x + 2)

2 x2 - x – 6 = 0 (x – 3)(x + 2) 3 x2 + 8x + 12 =

0(x + 2)(x + 6)

4 x2 + x – 12 = 0 (x – 3)(x + 4) 5 x2 - 8x + 15 =

0(x – 3)(x – 5)

6 x2 + 3x – 21 = 0

(x + 7)(x – 4)

7 x2 - 3x – 18 = 0

(x – 6)(x + 3)

8 x2 - 10x – 24 = 0

(x - 12)(x + 2)

9 x2 + 8x + 16 = 0

(x + 4)(x + 4)

10 x2 - 4x – 60 = 0

(x - 10)(x + 4)

6 and -3

3 and -2-2 and -63 and -43 and 5-7 and 4

-3 and -2

6 and -3

-4 and -4- 10 and -

4

FACTORING SPECIAL QUADRATIC EQUATIONS

THE DIFFERENCE BETWEEN

PERFECT SQUARES

FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES

(x2 + 0x - 4) Is This A Quadratic Equation?

Notice: x2 + 0x – 4 = (x2 – 4)

FACTORING (x2 + 0x - 4)

1 Find all the factor pairs of - 4 1 x -4 = -4

2 x -2 = -42 From these choose the pair that add up to “0” 2 + -2 = 0

3 Put these values into the brackets (x + _)(x + _) = 0 (x + 2)(x - 2) = 0


This is often called the “Difference between Two

Squares”x2 – 4

(x + 2)(x – 2)

This is often called the “Difference between Two

Squares”x2 – 4

(x + 2)(x – 2)


TO FACTOR THE DIFFERENCE BETWEEN SQUARES

2) MAKE THE BRACKETS { one (+) one (-) }

AND FILL IN THE BLANKS.

1) TAKE THE SQUARE ROOT OF THE BOTH TERMS .

x2 - 16

16

x2 – 16 = (x + 4 ) (x - 4 )

(x + 4 ) = 0 (x - 4 ) = 0

x = -4 x = 4

(x + __ ) (x - __ )4 4

x2 = 4= x

a + b

a+

b

a2 ab

ab b2

To Show Geometrically That

(a + b)2 = a2 + 2ab + b2

a2 +ab

+ab + b2

a2 + 2ab + b2

Now..

Cross Multiply

(a + b) (a + b)

a2 + 2ab + b2

To Show Algebraically That (a + b)2 = a2 + 2ab + b2

a b (a + b)(a + b)

a(a) ab+

+

+

x2 – 4

x2 + 0x – 4

(x – 2)(x + 2)

-1 x 4 = -4

-2 x 2 = -4

4 x -1 = -4

-2 + 2 = 0Notice that x2 – 4 could be written as

x2 – 22

(x – 2)(x + 2)

This is often called the difference between two

squaresx2 – 25

(x + 5)(x – 5)

This is often called the difference between two

squaresx2 – 25

(x + 5)(x – 5)

1 x2 - 9

2 x2 - 100

3 x2 - 36

4 x2 - 49

5 x2 - 81

(x + __ ) (x - __ )

1) MAKE THE BRACKETS { one (+) one (-) }

2) TAKE THE SQUARE ROOT OF THE NUMBER AND FILL IN THE BLANKS

3 3(x + 3) = 0 (x – 3) =

0

x = -3 x = 3

x = 3 or -3

USE YOUR WORKSHEET TO SOLVE THE DIFFERENCE OF SQUARES

1 x2 - 9 (x + 3)(x – 3)2 x2 - 100 (x + 10)(x – 10)3 x2 - 36 (x + 6)(x – 6)4 x2 - 49 (x + 7)(x – 7)5 x2 - 81 (x + 9)(x – 9)6 x2 - 64 (x + 8)(x – 8)7 x2 - 18 (x + √18)(x –

√18)8 x2 - 24 (x + √24)(x –

√24)

(x )(x )What goes with

the x?

(x )(x )What goes with

the x?

(x + 3)(x + 2)

x(x + 2) + 3(x + 2)

x X (x + 2) + 3 X (x + 2)

x X x + x X 2 + 3 X x + 3 X 2

x2 + 2x + 3x + 6

x2 + 5x + 6

You try

(x + 5)(x + 2)

(x – 2)(x + 3)

(x + 2)(x – 4)

(x – 3)(x – 2)

= (x + 3)(x + 4)

(x - 3) (x - 2)

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

x(x) + x(-2) - 3(x) - 3(-2)

x2 - 2x - 3x + 6

x2 - 5x + 6

PROOF:

x2 - 5x + 6 = (x - 3) (x - 2)

Factor:

Factor:

Combine like terms:

Documents

Solving Quadratic Equations by FACTORING What are Quadratic Equations? A quadratic equation is an equation which: y = x 2 y = x 2 + 2 y = x 2 + x – 4