A quadratic equation is written in the Standard Form, where a, b, and c are real numbers and. 5.8...

A quadratic equation is written in the Standard Form, 2 0ax bx c where a, b, and c are real numbers and .0a

5.8 – Solving Equations by Factoring

Zero Factor Property: 0ab

then or . 0a 0b If a and b are real numbers and if ,

Zero Factor Property: If a and b are real numbers and if , 0ab

Examples: 10 3 6 0x x

then or . 0a 0b

10 0x 3 6 0x

10x 3 6x 2x

10 10 01 0x 63 66 0x 3 6

3 3

x


Solving Equations by Factoring: 1) Write the equation to equal zero.

4) Solve each equation.

2) Factor the equation completely.

3) Set each factor equal to 0.

5) Check the solutions (in original equation).


2 3 18 0x x

6 0x 3 0x 3x

6x 3x

2 3 18x x

18 :Factors of1,18 2, 9 3, 6

26 3 16 8

36 18 18

18 18

213 3 83

9 9 18 18 18

6x 0


3 18x x 18x

2 3 18x x

218 13 18 8

324 54 18

270 18

221 23 11 8

441 63 18 378 18

3 18x

3 183 3x 21x

If the Zero Factor Property is not used, then the solutions will be incorrect


2 4 5x x

1 0x 5 0x

1 5 0x x

1x 5x

4 5x x

2 4 5 0x x


23 7 6x x 3 0x 3 2 0x

3 3 2 0x x

3x 2

3x

3 7 6x x

23 7 6 0x x 3 2x

6 :Factors of2, 31, 6

3:Factors of1, 3


29 24 16x x 29 24 16 0x x

3 4 0x 3 4 3 4 0x x

4

3x

3 4x

9 16and are perfect squares


32 18 0x x 2x

2 0x

2x

3x 3 0x 3 0x

3x 0x

2 9x 0

3x 3x 0


23 3 20 7 0x x x

3x

3 0x

7x 7 0x 3 1 0x

1

3x

3x 3 1x

3:Factors of 1, 3 7 :Factors of 1, 7

7x 0 3 1x


0

A cliff diver is 64 feet above the surface of the water. The formula for calculating the height (h) of the diver after t seconds is: 216 64.h t How long does it take for the diver to hit the surface of the water?

0 0

2 0t 2 0t 2t 2t seconds

216 64t 16 2 4t

16 2t 2t



2x

The square of a number minus twice the number is 63. Find the number.

7x

7x

x is the number.

2 2 63 0x x

7 0x 9 0x

9x

2x 63

63:Factors of 1, 63 3, 21 7, 9

9x 0

5 176w w

The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. What are the length and the width of the garden?

11w The width is w.

11 0w 11w

The length is w+5.l w A

2 5 176w w 2 5 176 0w w

16 0w 16w

11w 11 5l 16l

feet

feet

176 :Factors of1,176 2, 88 4, 44

8, 22 11,16

16w 0


x

Find two consecutive odd numbers whose product is 23 more than their sum?

Consecutive odd numbers: x

5x 5x 2 2 2 25x x x

2 25 0x 5x

5 0x 5 0x

5, 3 5, 7

5 2 3 5 2 7

2.x 2x 2x x 23

2 22 2 2 25xx x x x

2 25 2525x

2 25x

5x 0


a x

The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 meters. What are the lengths of the legs?

12a

.Pythagorean Th

22 27 13x x

5x

5

meters

7b x 13c

2 2 14 49 169x x x 22 14 120 0x x

22 7 60 0x x

2

5 0x 12 0x 12x

12 7b meters

2 2 2a b c

60 :Factors of 1, 60 2, 303, 20 4,15 5,12

5x 12x 0

6,10


6.1 – Rational Expressions -

A rational expression is a quotient of polynomials.

For any value or values of the variable that make the denominator zero, the rational expression is considered to be undefined at those value(s).

3

5 1

x

x

23 6 7

2 6

x x

x

4

3

4 4 12

q

q q The denominator can not equal zero.

Multiplying and Dividing

What are the values of the variable that make the denominator zero and the expression undefined?

3

5 1

x

x

23 6 7

2 6

x x

x

2

3 2

12

x

x x

5 1 0x

5 1x 1

5x

2 6 0x

2 6x 3x

3 4 0x x

2 12 0x x

3 0x 4 0x 3x 4x

6.1 – Rational Expressions – Mult. And Div.

Simplifying4 3

5 5

x x

x

2

5

25

q

q

3x

3

5

x

5q

1

5q

1x

5 1x 5q 5q


Simplifying2

2

11 18

2

x x

x x

2x

9

1

x

x

9x

2x 1x


Simplifying4

4

x

x

4x

4x

4x

1 4x 1


6.1 - Rational Expressions – Mult. And Div.

3 2

3

5 2

3 15

a b

b a

10

2

3a 2b45 a 3b

2a9 b

Multiplication:

Multiplication: 2

3 2

3 6 7

14 2

x x

x x

3

21

2x 2x

27x14 2x

1432


19

253

147

842

2

2

x

xx

xx

x

4 2x

7x 2x 3 1x

3 1x 3 1x 2x

4 2x

7x 3 1x

Multiplication:


y

xx

26

7 2

27

6

x 1

3

7xy

2y

x

1

3

Division:


2

123

6

4 2

xx

24

6

x

24

6

x

4

3

x 9

4x

2

3 12x

23 4x

1

1

3

1

3

Division:


2

25

4

410 23

2

x

xx

x

x

2

2

10 4

4

x

x

2

2x 2

22 xx

3 2

2

5 2

x

x x

5 2x

2x 2x 2x

2x 5 2x

2

1

x

Division:


21

129

147

8103 2

x

x

xx

21

21

23 10 8

7 14

x x

x

2x

1

7

21

9 12x

3 4x

7 2x 21

3 3 4x

21

31

Division:


6.2 – Rational ExpressionsAdding and Subtracting

6

5

3

2and

What is the Lowest Common Denominator (LCD)?

2

1

7

2,

4

3and

3 5

5 13and

x x4 2

17 3

4 6and

y y

6 28

5x 12 4y

5

7

5

3

a

aand

a

a


123

5

4

72

2

x

xand

x

x

5a

3

27xand 5x

4x 4x

243 x

4x 4x 3 4x 5a

6.2 – Rational ExpressionsAdding and Subtracting

23

3

32

522

yy

yand

yy

y


1y

5y

3y 2y

3y 1y and

3y

1y 2y

6.2 – Rational Expressions – Add. And Sub.

73

7

73

3

xx

x

Examples (Like Denominators):

1

3 7x 3x 7


2

64

2

52 2

x

x

x

xx


2x 22 5

2

x x

x

22

2

x

x

2x

32 x22 5x x 4 6x

4x 6

x 6

2x 2 3x


65

72

65

1322

xx

x

xx

x


65

72132

xx

xx

65

132

xx

x x2 7

65

62

xx

x

6x

x 1 x 6

1

1

x

1

1


Examples:

15

3

5

yy

LCD 15

15

3

5

yy

3

3

15

3

15

3 yy

15

33 yy 15

00


Examples:

210

11

8

5

xx

LCD 40x2

210

11

8

5

xx

x

x

5

5

22 40

44

40

25

xx

x

240

1125

x

x

4

4


Examples:

1

2

7

5

xx

LCD

5 2

7 1x x

7 1 7 1x x x x

7 1x x

7x 1x

1

1

x

x

7

7

x

x

5x 5 14x 19x 5


3

5

9

102

xx

xExamples:

10x

10 5

3 3 3

x

x x x

10

3 3

x

x x

3 3x x

3 3x x

3 3x x

3

5

x

3x 3x

LCD

3

3

x

x

3 3x x 5x 15

10x 155x

5x 15

5 3x 3x 3x

5

3x


812

3

23

42

x

x

xx

Examples:

LCD

4

4 3

3 2 4 3 2

x

x x x

x 3 2x 3x

4 3 2x

4 x 3 2x

4

4

x

x


Examples: continued

4 3 2 4 3 2x x x x

4 3 2x x

x

x

x

x

xx 234

3

23

4

4

4

16 23x

216 3x


444

622

x

x

xx

xExamples:

6x x

LCD

6

2 2 2 2

x x

x x x x

2x 2x 2x 2x

2x 2x 2x

2

2

x

x

2

2

x

x


Examples: continued

2

2

2222

6

2

2

x

x

xx

x

xx

x

x

x

2 2 2 2 2 2x x x x x x

222

2126 22

xxx

xxxx

2 2 2x x x 22 2x x

26x 12x2x 2x

27x 10x x 7x 10


6.3 – Rational ExpressionsSimplifying Complex Fractions

LCD:

3759

3759

3 5

7 9

6337

63593 9

7 5

9 3

7 5

27

3527

35

63

Outers over Inners

Simplifying Complex Fractions

3 24 31 32 8

3 24 31 32 8

3 43 4

4

3 24 3

1 384 2

24 24

2

3 24 31

43

2 824

9 812 12

4 38 8

6 3 8 2

12 1 3 3

11278

18 16

12 9

2

211 8

12 7

1 8 2

12 3 7

2

21

LCD: 12, 8 LCD: 24

6.3 – Rational Expressions


LCD: y

1

2 1

xy

xy

1

2 1

y y

y

xy

xy

2 1

y x

x

yx

yx

12

1y–y



LCD: 6xy

56

3

yy xyx

2

2 2

5 6

2 6

x y

xy x y

56

3

6 6

6 6

yy x

xy xy

yy

xx xy

25 6x y2xy 3y x

xy

xy

y

3

656xy

6xy


END

Date post:	02-Jan-2016
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A quadratic equation is written in the Standard Form, where a, b, and c are real numbers and. 5.8...

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