Question 201048: Factor completely.
x2 + 3xy - 208y2
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 1 and -208 respectively.
Now multiply the first coefficient 1 and the last coefficient -208 to get -208. Now what two numbers multiply to -208 and add to the middle coefficient 3? Let's list all of the factors of -208:
Factors of -208:
1,2,4,8,13,16,26,52,104,208
-1,-2,-4,-8,-13,-16,-26,-52,-104,-208 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -208
(1)*(-208)
(2)*(-104)
(4)*(-52)
(8)*(-26)
(13)*(-16)
(-1)*(208)
(-2)*(104)
(-4)*(52)
(-8)*(26)
(-13)*(16)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3
| First Number | Second Number | Sum | | 1 | -208 | 1+(-208)=-207 | | 2 | -104 | 2+(-104)=-102 | | 4 | -52 | 4+(-52)=-48 | | 8 | -26 | 8+(-26)=-18 | | 13 | -16 | 13+(-16)=-3 | | -1 | 208 | -1+208=207 | | -2 | 104 | -2+104=102 | | -4 | 52 | -4+52=48 | | -8 | 26 | -8+26=18 | | -13 | 16 | -13+16=3 |
From this list we can see that -13 and 16 add up to 3 and multiply to -208
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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