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Start with the given expression
Rearrange the terms
Now let
. So our equation becomes
Notice how I replaced each
with z
Looking at
we can see that the first term is
and the last term is
where the coefficients are 1 and 1 respectively.
Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient -2? Let's list all of the factors of 1:
Factors of 1:
1
-1 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 1
1*1
(-1)*(-1)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2
| First Number | Second Number | Sum | | 1 | 1 | 1+1=2 |
| -1 | -1 | -1+(-1)=-2 |
From this list we can see that -1 and -1 add up to -2 and multiply to 1
Now looking at the expression
, replace
with
(notice
adds up to
. So it is equivalent to
)
Now let's factor
by grouping:
Group like terms
Factor out the GCF of
out of the first group. Factor out the GCF of
out of the second group
Since we have a common term of
, we can combine like terms
So
factors to
So this also means that
factors to
(since
is equivalent to
)
note:
is equivalent to
since the term
occurs twice. So
also factors to
So
factors to
Remember, we let
. So replace each z with
Combine like terms
Square -y to get
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Answer:
So
factors to
(