Question 168159: 3x^2+37xy-86y^2
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 3 and -86 respectively.
Now multiply the first coefficient 3 and the last coefficient -86 to get -258. Now what two numbers multiply to -258 and add to the middle coefficient 37? Let's list all of the factors of -258:
Factors of -258:
1,2,3,6,43,86,129,258
-1,-2,-3,-6,-43,-86,-129,-258 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -258
(1)*(-258)
(2)*(-129)
(3)*(-86)
(6)*(-43)
(-1)*(258)
(-2)*(129)
(-3)*(86)
(-6)*(43)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 37? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 37
| First Number | Second Number | Sum | | 1 | -258 | 1+(-258)=-257 | | 2 | -129 | 2+(-129)=-127 | | 3 | -86 | 3+(-86)=-83 | | 6 | -43 | 6+(-43)=-37 | | -1 | 258 | -1+258=257 | | -2 | 129 | -2+129=127 | | -3 | 86 | -3+86=83 | | -6 | 43 | -6+43=37 |
From this list we can see that -6 and 43 add up to 37 and multiply to -258
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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