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SOLUTION: Factor the following expression completely. x^2 - 8x + 16 - 9y^2
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Question 318020
:
Factor the following expression completely.
x^2 - 8x + 16 - 9y^2
Answer by
jim_thompson5910(33401)
(
Show Source
):
You can
put this solution on YOUR website!
First let's focus on
Looking at the expression
, we can see that the first coefficient is
, the second coefficient is
, and the last term is
.
Now multiply the first coefficient
by the last term
to get
.
Now the question is: what two whole numbers multiply to
(the previous product)
and
add to the second coefficient
?
To find these two numbers, we need to list
all
of the factors of
(the previous product).
Factors of
:
1,2,4,8,16
-1,-2,-4,-8,-16
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to
.
1*16 = 16
2*8 = 16
4*4 = 16
(-1)*(-16) = 16
(-2)*(-8) = 16
(-4)*(-4) = 16
Now let's add up each pair of factors to see if one pair adds to the middle coefficient
:
First Number
Second Number
Sum
1
16
1+16=17
2
8
2+8=10
4
4
4+4=8
-1
-16
-1+(-16)=-17
-2
-8
-2+(-8)=-10
-4
-4
-4+(-4)=-8
From the table, we can see that the two numbers
and
add to
(the middle coefficient).
So the two numbers
and
both multiply to
and
add to
Now replace the middle term
with
. Remember,
and
add to
. So this shows us that
.
Replace the second term
with
.
Group the terms into two pairs.
Factor out the GCF
from the first group.
Factor out
from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.
Combine like terms. Or factor out the common term
Condense the terms.
So
factors to
.
In other words,
.
----------------------------------------------------------
So
Now let's factor
Start with the given expression.
Rewrite
as
.
Notice how we have a difference of squares
where in this case
and
.
So let's use the difference of squares formula
to factor the expression:
Start with the difference of squares formula.
Plug in
and
.
So this shows us that
factors to
.
So this then means that
.