Question 145894: Factor completely:
7x^2 + 58x + 16
Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628)   (Show Source):
Answer by jim_thompson5910(35256)  (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 7 and 16 respectively.
Now multiply the first coefficient 7 and the last coefficient 16 to get 112. Now what two numbers multiply to 112 and add to the middle coefficient 58? Let's list all of the factors of 112:
Factors of 112:
1,2,4,7,8,14,16,28,56,112
-1,-2,-4,-7,-8,-14,-16,-28,-56,-112 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 112
1*112
2*56
4*28
7*16
8*14
(-1)*(-112)
(-2)*(-56)
(-4)*(-28)
(-7)*(-16)
(-8)*(-14)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 58? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 58
| First Number | Second Number | Sum | | 1 | 112 | 1+112=113 | | 2 | 56 | 2+56=58 | | 4 | 28 | 4+28=32 | | 7 | 16 | 7+16=23 | | 8 | 14 | 8+14=22 | | -1 | -112 | -1+(-112)=-113 | | -2 | -56 | -2+(-56)=-58 | | -4 | -28 | -4+(-28)=-32 | | -7 | -16 | -7+(-16)=-23 | | -8 | -14 | -8+(-14)=-22 |
From this list we can see that 2 and 56 add up to 58 and multiply to 112
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
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