Download AIM 05-7-S Solving Quadratic Equations by Factoring FINAL KM

Solving Equations
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Factors that multiply
to Zero ... ab=0
That’s
Whatright!
if
ab = 0equal
?
a or b could
zero!
Then
a could
equal zero!
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...or b
could equal
zero!
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ZERO Product Rule
If a a
product
 b is zero,
0
then
one or or
morebofthe
0
a factors
0
must be
zero!
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Zero Factor Property
Solve: 2x = 0
2x  0
20
x0
0
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Zero Factor Property
Solve: 2(x + 5) = 0
2x  5   0
20
x5 0
x  5
 5
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Zero Factor Property
Solve: 2x(x + 5) = 0
2xx  5   0
20 x 0 x5 0
x  5
0,5
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Zero Factor Property
Solve: 2x(x + 5)(x-3) = 0
2xx  5 x  3   0
2  0 x  0 x5  0 x3  0
x  5 x  3
0,5,3
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Zero Factor Property
Solve: (2x - 7)(4x + 3)= 0
2x  74x  3  0
2x  7  0
4x  3  0
4x  3
3
x
4
2x  7
7
x
2
7 3 
 , 
2 4 
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Zero Factor Property
Solve: 6x2 = 3x
Use the properties of equality to
rearrange the terms of the equation so
that it is equal to ZERO.
6x  3x
2
6x  3x  0
3x2x  1  0
2
30
x0
2x  1  0
2x  1
 1
0, 
 2
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x
2
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General Strategy for
Solving Equations Using
The Zero Factor Property
1) Arrange the equation so that one
side is zero.
2) Completely factor the other side.
3) Set each factor equal to zero
and solve, if possible.
4) Write the solution set.
5) Check each solution by substitution.
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Solve: x2 = 169
1) Arrange the equation so that one
side is zero.
x  169  0
2
2) Completely factor the expression.
(x  13)(x  13)  0
3) Set each factor equal to zero and solve.
x  13  0
x  13
or
or
x  13  0
x  13
4) Write the solution set.
 13,13
5) Check each solution by substitution.
( 13)2  169
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(13)2  169
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Solve: x2 + 25 = 10x
1) Arrange the equation so that one
side is zero.
x  10x  25  0
2
2) Completely factor the expression.
(x  5)(x  5)  0
3) Set each factor equal to zero and solve.
x  5  0 or x  5  0
x5
or
x5
4) Write the solution set.
5
5) Check each solution by substitution.
(5)2  25  105 50  50
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Solve: 3x2 = 2 - x
1) Arrange the equation so that one
side is zero.
3x  x  2  0
2
2) Completely factor the expression.
(3x  2)(x  1)  0
3) Set each factor equal to zero and solve.
3x  2  0
2
x
3
or
x 1 0
or
x  1
4) Write the solution set.
2

 ,1
3

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Solve: 3x2 = 2 – x
continued
2

 ,1
3

5) Check each solution by substitution.
2
2
2
3   2   
3
3
4 6 2

3     
9 3 3
3 1  2   1
2
31  2  1
33
4 4

3 3
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Solve: x + 12 = x(x – 3)
1) Arrange the equation so that one
side is zero.
x  12  x 2  3x
x  4x  12  0
2
2) Completely factor the expression.
(x  6)(x  2)  0
3) Set each factor equal to zero and solve.
x  6  0 or x  2  0
x6
or
x  2
4) Write the solution set.
6,2
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Solve: x + 12 = x(x – 3)
continued
6,2
5) Check each solution by substitution.
6  12  66  3
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 2  12  2(2  3)
18  6(3)
10  2 5 
18  18
10  10
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Solve:
2x3 + 3x2 = 18x + 27
1) Arrange the equation so that one
side is zero.
2x  3x  18x  27  0
3
2
2) Completely factor the expression.
x (2x  3)  9(2x  3)  0
2
(x  9)(2x  3)  0
2
(x  3)(x  3)(2x  3)  0
3) Set each factor equal to zero and solve.
x  3  0 x  3  0 2x  3  0
x  3
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or
x3
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or
3
x
2
17
Solve:
2x3 + 3x2 = 18x + 27
continued
4) Write the solution set.
3

 3,3, 
2

5) Check each solution by substitution.
2 3  3 3  18 3  27
3
2
2 27   39   54  27
 54  27  27
 27  27
233  332  183  27
227   39   54  27
54  27  81
81 81
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Solve:
2x3 + 3x2 = 18x + 27
continued
4) Write the solution set.
3

 3,3, 
2

5) Check each solution by substitution.
3
2
 3
 3
 3
2    3    18    27
 2
 2
 2
 27 
9
2 
  3   27  27
 8 
4
27 27


0
4
4
00
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That’s All for Now!
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