Factoring Worksheet With Examples Page 36
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50Proofs
When you solve an equation by factoring, you are using a deductive argument.
Each step can be justified by an algebraic property.
2
Solve 4x
- 324 = 0.
2
4x
- 324 = 0
Original equation
2
2
2
2
2
(2x)
- 18
= 0
4x
= (2x)
and 324 = 18
(2x + 18)(2x - 18) = 0
Factor the difference of squares.
2x + 18 = 0 or 2x - 18 = 0
Zero Product Property
x = -9
x = 9
Solve each equation.
Notice that the column on the left is a step-by-step process that leads to a
solution. The column on the right contains the reasons for each statement.
A two-column proof is a deductive argument that contains statements and
reasons.
Two-Column Proof
Given: a, x, and y are real numbers such that a ≠ 0, x ≠ 0, and y ≠ 0.
There is a reason for
4
4
2
2
Prove: ax
- ay
= a( x
+ y
)(x + y)(x - y)
each statement.
Statements
Reasons
The first statement
1. a, x, and y are real numbers such that
1. Given
contains the given
a ≠ 0, x ≠ 0, and y ≠ 0.
information.
4
4
4
4
4
4
2. ax
- ay
= a( x
- y
)
2. The GCF of ax
and ay
is a.
2
2
2
2
4
4
2
2
4
2
4
2
3. ax
3. x
- ay
= a[ ( x
)
- ( y
)
]
= ( x
)
and y
= ( y
)
The last statement
4
4
2
2
2
2
4. ax
- ay
= a( x
+ y
)( x
- y
)
4. Factor the difference of squares.
is what you want
4
4
2
2
to prove.
5. ax
- ay
= a( x
+ y
)(x + y)(x - y)
5. Factor the difference of squares.
Reading to Learn
_
1
2
1. Solve
- 100 = 0 by using a two-column proof.
t
16
2. Write a two-column proof using the following information. (Hint: Group terms with
common factors.)
Given: c and d are real numbers such that c ≠ 0 and d ≠ 0.
3
2
2
3
Prove: c
- cd
- c
d + d
= (c + d)(c - d)(c - d)
3. Explain how the process used to write two-column proofs can be useful in solving
Find the Error exercises, such as Exercise 37 on page 451.
453
Reading Math Proofs
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