Question 352211: Factor completely.
20w^2+100wg+125g^2
All of the examples I have offer numbers that have square roots so i'm not sure how to solve this one. Any help you can give is very much appreciated.
Thank you in advance!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
 Start with the given expression
 Factor out the GCF 
Now let's focus on the inner expression 
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Looking at we can see that the first term is  and the last term is  where the coefficients are 4 and 25 respectively.
Now multiply the first coefficient 4 and the last coefficient 25 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:
Factors of 100:
1,2,4,5,10,20,25,50
-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20
| First Number | Second Number | Sum | | 1 | 100 | 1+100=101 | | 2 | 50 | 2+50=52 | | 4 | 25 | 4+25=29 | | 5 | 20 | 5+20=25 | | 10 | 10 | 10+10=20 | | -1 | -100 | -1+(-100)=-101 | | -2 | -50 | -2+(-50)=-52 | | -4 | -25 | -4+(-25)=-29 | | -5 | -20 | -5+(-20)=-25 | | -10 | -10 | -10+(-10)=-20 |
From this list we can see that 10 and 10 add up to 20 and multiply to 100
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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So our expression goes from and factors further to 
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Answer:
So factors to
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