a quadratic equation is written in the standard form ,
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5.8 – Solving Equations by Factoring. A quadratic equation is written in the Standard Form ,. where a , b , and c are real numbers and. Zero Factor Property:. If a and b are real numbers and if ,. then or. 5.8 – Solving Equations by Factoring. - PowerPoint PPT PresentationTRANSCRIPT
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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 ,
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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
5.8 – Solving Equations by Factoring
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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).
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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2 4 5x x
1 0x 5 0x
1 5 0x x
1x 5x
4 5x x
2 4 5 0x x
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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32 18 0x x 2x
2 0x
2x
3x 3 0x 3 0x
3x 0x
2 9x 0
3x 3x 0
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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5.8 – Solving Equations by Factoring
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
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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
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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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
5.8 – Solving Equations by Factoring
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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
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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.
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Simplifying4 3
5 5
x x
x
2
5
25
q
q
3x
3
5
x
5q
1
5q
1x
5 1x 5q 5q
6.1 – Rational Expressions – Mult. And Div.
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Simplifying2
2
11 18
2
x x
x x
2x
9
1
x
x
9x
2x 1x
6.1 – Rational Expressions – Mult. And Div.
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Simplifying4
4
x
x
4x
4x
4x
1 4x 1
6.1 – Rational Expressions – Mult. And Div.
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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:
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Multiplication: 2
3 2
3 6 7
14 2
x x
x x
3
21
2x 2x
27x14 2x
1432
6.1 - Rational Expressions – Mult. And Div.
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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:
6.1 - Rational Expressions – Mult. And Div.
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y
xx
26
7 2
27
6
x 1
3
7xy
2y
x
1
3
Division:
6.1 - Rational Expressions – Mult. And Div.
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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:
6.1 - Rational Expressions – Mult. And Div.
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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:
6.1 - Rational Expressions – Mult. And Div.
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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:
6.1 - Rational Expressions – Mult. And Div.
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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
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5
7
5
3
a
aand
a
a
What is the Lowest Common Denominator (LCD)?
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
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23
3
32
522
yy
yand
yy
y
What is the Lowest Common Denominator (LCD)?
1y
5y
3y 2y
3y 1y and
3y
1y 2y
6.2 – Rational Expressions – Add. And Sub.
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73
7
73
3
xx
x
Examples (Like Denominators):
1
3 7x 3x 7
6.2 – Rational Expressions – Add. And Sub.
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2
64
2
52 2
x
x
x
xx
Examples (Like Denominators):
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
6.2 – Rational Expressions – Add. And Sub.
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65
72
65
1322
xx
x
xx
x
Examples (Like Denominators):
65
72132
xx
xx
65
132
xx
x x2 7
65
62
xx
x
6x
x 1 x 6
1
1
x
1
1
6.2 – Rational Expressions – Add. And Sub.
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Examples:
15
3
5
yy
LCD 15
15
3
5
yy
3
3
15
3
15
3 yy
15
33 yy 15
00
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
6.2 – Rational Expressions – Add. And Sub.
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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
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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
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Simplifying Complex Fractions
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
6.3 – Rational Expressions
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Simplifying Complex Fractions
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
6.3 – Rational Expressions
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