Question 154208: Factor: 25v^2 + 30vw + 9w^2
please explain thoroughly as I would like to know how to solve on my own. thanks
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 25 and 9 respectively.
Now multiply the first coefficient 25 and the last coefficient 9 to get 225. Now what two numbers multiply to 225 and add to the middle coefficient 30? Let's list all of the factors of 225:
Factors of 225:
1,3,5,9,15,25,45,75
-1,-3,-5,-9,-15,-25,-45,-75 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 225
1*225
3*75
5*45
9*25
15*15
(-1)*(-225)
(-3)*(-75)
(-5)*(-45)
(-9)*(-25)
(-15)*(-15)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 30? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 30
| First Number | Second Number | Sum | | 1 | 225 | 1+225=226 | | 3 | 75 | 3+75=78 | | 5 | 45 | 5+45=50 | | 9 | 25 | 9+25=34 | | 15 | 15 | 15+15=30 | | -1 | -225 | -1+(-225)=-226 | | -3 | -75 | -3+(-75)=-78 | | -5 | -45 | -5+(-45)=-50 | | -9 | -25 | -9+(-25)=-34 | | -15 | -15 | -15+(-15)=-30 |
From this list we can see that 15 and 15 add up to 30 and multiply to 225
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 )
note: is equivalent to  since the term occurs twice. So also factors to
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Answer:
So factors to
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