Solving Quadratic Equations by FACTORING

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SolvingQuadratic Equationsby FACTORING
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What are Quadratic Equations?
A quadratic equation is an equation which y
x2 y x2 2 y x2 x 4 y
x2 2x 3
Contains a x2 term
All of these equations contain a x2 term
therefore they are called
Quadratic Equations
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Which of the following are Quadratic Equations?
y x 3
It Contains a x2 term
y x2
....WHY?
y 2x 4
It Contains a x2 term
y x2 2x 3
.WHY?
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Solving Quadratic Equations BY FACTORING
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
1 x 6 and 2 x 3
The () factors of 6 are
-1 x -6 and -2 x -3
The (-) factors of 6 are
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Solving Quadratic Equations BY FACTORING
BIG IDEA NUMBER ONE
If A(B) 0
what can we say about
either A or B?
Either A or B must equal ZERO!!!
A 0 or B 0
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Solving Quadratic Equations BY FACTORING
BIG IDEA NUMBER ONE
So if
(x 3) (x 3) 0
THEN EITHER (x 3) 0 or (x 3) 0
So.
x -3 or x 3
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Solving Quadratic Equations 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

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How to solve Quadratic Equations by FACTORING
Example 1
x2 x 0
7
7
12
12
1 x 12 12 -1 x -12 12 2 x 6 12 -2 x -6
12 3 x 4 12 -3 x -4 12
Write down all the factor pairs of ___.

• (x )(x ) 0
• What goes with the x?

1
Positive Negative
From this list, choose the pair that adds up to
___
2
3 4 7
0 (x )(x )
0 (x 3)(x 4)
0 (x 3)(x 4) x 3 and 4
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Put these numbers into brackets
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PROOF x2 7x 12 (x 3) (x 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
Factor
Factor
Combine like terms
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How to solve Quadratic Equations by FACTORING
Example 2 x2 x
0
- 5
- 5
6
6
1 x 6 6 -1 x -6 6 2 x 3 6 -2 x -3 6
Write down all the factor pairs of ___ .
1
Positive Negative
From this list, choose the pair that adds up to
___ .
2
-2 -3 -5
(x - 2)(x - 3) O x 2 and 3
3
Put these numbers into brackets
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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
1
From this list, choose the pair that adds up to 1
2

(3) (-2) 1 3 2 1
0 (x 3)(x - 2) x 3 and 2

Put these numbers into brackets
3
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USE WORKSHEET 1 USE WORKSHEET 1
x2 3x 2 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
x2 x 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
x2 12x 20 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
.
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
2
1 x 2 2 -1 x -2 2
PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS
POWERPOINT WORKSHEET 1
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1 2 3
2
1
12
1 x 12 12 -1 x -12 12 2 x 6 12 -2 x -6
12 3 x 4 12 -3 x -4 12
WORK TOGETHER TO FACTOR THE NEXT QUADRATIC
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2 6 8
2
6
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1 x2 5x 6 0
2 x2 - x 6 0
3 x2 8x 12 0
4 x2 x 12 0
5 x2 - 8x 15 0
6 x2 3x 28 0
7 x2 - 3x 18 0
8 x2 - 10x 24 0
9 x2 8x 16 0
10 x2 - 6x 40 0
(x 3)(x 2)
(x 3)(x 2) (x 2)(x 6) (x 3)(x 4) (x
3)(x 5) (x 7)(x 4)
PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS
POWERPOINT WORKSHEET 2
(x 6)(x 3)
(x - 12)(x 2)
(x 4)(x 4)
(x - 10)(x 4)
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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)
-3 and -2
3 and -2
-2 and -6
3 and -4
3 and 5
-7 and 4
6 and -3
6 and -3
-4 and -4
- 10 and -4
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FACTORING SPECIAL QUADRATIC EQUATIONS

• THE DIFFERENCE BETWEEN

PERFECT SQUARES
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FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES

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

FACTORING (x2 0x - 4)
1 x -4 -4 2 x -2 -4
1 Find all the factor pairs of - 4
2 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
Notice x2 0x 4 (x2 4)
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FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES
This is often called the Difference between Two
Squares x2 4 (x 2)(x 2)
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FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES
TO FACTOR THE DIFFERENCE BETWEEN SQUARES
x2 - 16
1) TAKE THE SQUARE ROOT OF THE BOTH TERMS .
x2
16
4
x
2) MAKE THE BRACKETS one () one (-)
AND FILL IN THE BLANKS.
(x __ ) (x - __ )
4
4
x2 16 (x 4 ) (x - 4 )
(x 4 ) 0 (x - 4 ) 0 x
-4 x 4
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To Show Geometrically That (a b)2 a2 2ab
b2
Now.. Cross Multiply
a2
ab
b2
ab
a2 2ab b2
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To Show Algebraically That (a b)2 a2 2ab
b2

• (a b) (a b)
• a2 2ab b2

a
b
(a b)
(a b)

a(a)
ab


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-1 x 4 -4 -2 x 2 -4 4 x -1 -4 -2 2 0
x2 4 x2 0x 4 (x 2)(x 2)
Notice that x2 4 could be written as x2 22 (x
2)(x 2)
This is often called the difference between two
squares x2 25 (x 5)(x 5)
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USE YOUR WORKSHEET TO SOLVE THE DIFFERENCE OF
SQUARES
1) MAKE THE BRACKETS one () one (-)
2) TAKE THE SQUARE ROOT OF THE NUMBER AND FILL IN
THE BLANKS
1 x2 - 9
2 x2 - 100
3 x2 - 36
4 x2 - 49
5 x2 - 81



(x 3) 0 (x 3) 0 x -3 x 3 x 3 or -3
(x __ ) (x - __ )
3
3
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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 v18)(x v18)
8 x2 - 24 (x v24)(x v24)
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• (x )(x )
• What goes with the x?

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(x 3)(x 2)
(x 3)(x 4)
You try (x 5)(x 2) (x 2)(x 3) (x 2)(x
4) (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
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PROOF x2 - 5x 6 (x - 3) (x - 2)
(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
Factor
Factor
Combine like terms