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Question 282701: factor 4x^2 + 7xy - 15y^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 4 and -15 respectively.
Now multiply the first coefficient 4 and the last coefficient -15 to get -60. Now what two numbers multiply to -60 and add to the middle coefficient 7? Let's list all of the factors of -60:
Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -60
(1)*(-60)
(2)*(-30)
(3)*(-20)
(4)*(-15)
(5)*(-12)
(6)*(-10)
(-1)*(60)
(-2)*(30)
(-3)*(20)
(-4)*(15)
(-5)*(12)
(-6)*(10)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7
| First Number | Second Number | Sum | | 1 | -60 | 1+(-60)=-59 | | 2 | -30 | 2+(-30)=-28 | | 3 | -20 | 3+(-20)=-17 | | 4 | -15 | 4+(-15)=-11 | | 5 | -12 | 5+(-12)=-7 | | 6 | -10 | 6+(-10)=-4 | | -1 | 60 | -1+60=59 | | -2 | 30 | -2+30=28 | | -3 | 20 | -3+20=17 | | -4 | 15 | -4+15=11 | | -5 | 12 | -5+12=7 | | -6 | 10 | -6+10=4 |
From this list we can see that -5 and 12 add up to 7 and multiply to -60
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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