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Start with the given equation
Rearrange the terms
Looking at
we can see that the first term is
and the last term is
where the coefficients are 1 and 56 respectively.
Now multiply the first coefficient 1 and the last coefficient 56 to get 56. Now what two numbers multiply to 56 and add to the middle coefficient -15? Let's list all of the factors of 56:
Factors of 56:
1,2,4,7,8,14,28,56
-1,-2,-4,-7,-8,-14,-28,-56 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 56
1*56
2*28
4*14
7*8
(-1)*(-56)
(-2)*(-28)
(-4)*(-14)
(-7)*(-8)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -15
| First Number | Second Number | Sum | | 1 | 56 | 1+56=57 |
| 2 | 28 | 2+28=30 |
| 4 | 14 | 4+14=18 |
| 7 | 8 | 7+8=15 |
| -1 | -56 | -1+(-56)=-57 |
| -2 | -28 | -2+(-28)=-30 |
| -4 | -14 | -4+(-14)=-18 |
| -7 | -8 | -7+(-8)=-15 |
From this list we can see that -7 and -8 add up to -15 and multiply to 56
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
)
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Answer:
So
factors to
(