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Looking at
we can see that the first term is
and the last term is
where the coefficients are 1 and 49 respectively.
Now multiply the first coefficient 1 and the last coefficient 49 to get 49. Now what two numbers multiply to 49 and add to the middle coefficient -14? Let's list all of the factors of 49:
Factors of 49:
1,7
-1,-7 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 49
1*49
7*7
(-1)*(-49)
(-7)*(-7)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -14? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -14
| First Number | Second Number | Sum | | 1 | 49 | 1+49=50 |
| 7 | 7 | 7+7=14 |
| -1 | -49 | -1+(-49)=-50 |
| -7 | -7 | -7+(-7)=-14 |
From this list we can see that -7 and -7 add up to -14 and multiply to 49
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
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Answer:
So
factors to
(