1. Solving Quadratic Equations by FACTORING

2. 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

3. 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?

4. 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:
The (-) factors of 6 are:
-1 x -6 and -2 x -3

5. BIG IDEA NUMBER ONE Solving Quadratic Equations BY FACTORING
If A(B) = what can we say about either A or B?
Either A or B must equal ZERO!!!
A = 0 or B = 0

6. 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 = or x = 3

7. BIG IDEA NUMBER ONE Solving Quadratic Equations BY FACTORING
TO SOLVE A QUADRATIC EQUATION BY FACTORING
MAKE THE EQUATION EQUAL TO ZERO
FACTOR THE EQUATION
SET THE FACTORS EQUAL TO ZERO AND SOLVE

8. How to solve Quadratic Equations by FACTORING
Example x2 + x + = 0 7 7 12 12
1 x 12 = x -12 = 12
2 x 6 = x -6 = 12
3 x 4 = 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 + 3)(x + 4)
x = – 3 and – 4
0 = (x + )(x + )
0 = (x + 3)(x + 4) 3
Put these numbers into brackets

9. (x + 3) (x + 4) x(x + 4) + 3(x + 4) x(x) + x(4) + 3(x) + 3(4)
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:

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

11. Solve by factoring: x2 + x - 6 = 0
Write down all the factor pairs of – 6
1 x -6 = -6
2 x -3 = -6
3 x -2 = -6
6 x -1 = -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

12. PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT
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
From these choose the pair that add up to ­_____
(x + _)(x + _) = 0
x2 – 12x – 20 = 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 = x -2 = 2
PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT
WORKSHEET # 1 3
1 + 2= 3
+ 1
+ 2 12
1 x 12 = x -12 = 12
2 x 6 = x -6 = 12
3 x 4 = x -4 = 12
WORK TOGETHER TO FACTOR THE NEXT QUADRATIC 8
2 + 6 = 8 +2 +6

13. PLEASE TAKE OUT YOUR QUADRATIC EQUATIONS POWERPOINT1
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)

14. 1 x2 + 5x + 6 = 0 (x + 3)(x + 2) 2 x2 - x – 6 = 0 (x – 3)(x + 2) 34
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

15. FACTORING SPECIAL QUADRATIC EQUATIONS
THE DIFFERENCE BETWEEN
PERFECT SQUARES

16. 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”
= 0
3 Put these values into the brackets (x + _)(x + _) = 0
(x + 2)(x - 2) = 0
Notice:
x2 + 0x – 4 = (x2 – 4)

17. FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES
This is often called the “Difference between Two Squares”
x2 – 4
(x + 2)(x – 2)

18. FACTORING THE DIFFERENCE BETWEEN PERFECT SQUARES
TO FACTOR THE DIFFERENCE BETWEEN SQUARES
x2 - 16
1) TAKE THE SQUARE ROOT OF THE BOTH TERMS . x2
= x 16
= 4
2) MAKE THE BRACKETS { one (+) one (-) }
AND FILL IN THE BLANKS.
(x + __ ) (x - __ ) 4 4
x2 – 16 = (x + 4 ) (x - 4 )
(x + 4 ) = (x - 4 ) = 0
x = x = 4

19. To Show Geometrically That
(a + b)2 = a2 + 2ab + b2 a
+ b
Now..
Cross Multiply a a2 ab a2
+ab + b ab b2
+ b2
+ab
a2 + 2ab + b2

20. 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 + +

21. This is often called the difference between two squares
-1 x 4 = -4
-2 x 2 = -4
4 x -1 = -4
= 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)

22. 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 x 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

23. 1
x2 - 9
(x + 3)(x – 3) 2 x
(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)

24. (x )(x )
What goes with the x?

25. (x + 3)(x + 2) x(x + 2) + 3(x + 2)  x X (x + 2) + 3 X (x + 2)
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 X x + 3 X 2
 x2 + 2x + 3x + 6
 x2 + 5x + 6

26. (x - 3) (x - 2) x(x - 2) -3(x - 2) x(x) + x(-2) - 3(x) - 3(-2)
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: