SOLUTION: How do I factor 18x^2-60xy+50y^2

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Question 542152: How do I factor 18x^2-60xy+50y^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

18x%5E2-60xy%2B50y%5E2 Start with the given expression.


2%289x%5E2-30xy%2B25y%5E2%29 Factor out the GCF 2.


Now let's try to factor the inner expression 9x%5E2-30xy%2B25y%5E2


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Looking at the expression 9x%5E2-30xy%2B25y%5E2, we can see that the first coefficient is 9, the second coefficient is -30, and the last coefficient is 25.


Now multiply the first coefficient 9 by the last coefficient 25 to get %289%29%2825%29=225.


Now the question is: what two whole numbers multiply to 225 (the previous product) and add to the second coefficient -30?


To find these two numbers, we need to list all of the factors of 225 (the previous product).


Factors of 225:
1,3,5,9,15,25,45,75,225
-1,-3,-5,-9,-15,-25,-45,-75,-225


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 225.
1*225 = 225
3*75 = 225
5*45 = 225
9*25 = 225
15*15 = 225
(-1)*(-225) = 225
(-3)*(-75) = 225
(-5)*(-45) = 225
(-9)*(-25) = 225
(-15)*(-15) = 225

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -30:


First NumberSecond NumberSum
12251+225=226
3753+75=78
5455+45=50
9259+25=34
151515+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 the table, we can see that the two numbers -15 and -15 add to -30 (the middle coefficient).


So the two numbers -15 and -15 both multiply to 225 and add to -30


Now replace the middle term -30xy with -15xy-15xy. Remember, -15 and -15 add to -30. So this shows us that -15xy-15xy=-30xy.


9x%5E2%2Bhighlight%28-15xy-15xy%29%2B25y%5E2 Replace the second term -30xy with -15xy-15xy.


%289x%5E2-15xy%29%2B%28-15xy%2B25y%5E2%29 Group the terms into two pairs.


3x%283x-5y%29%2B%28-15xy%2B25y%5E2%29 Factor out the GCF 3x from the first group.


3x%283x-5y%29-5y%283x-5y%29 Factor out the -5y from the second group.


%283x-5y%29%283x-5y%29 Factor out 3x-5y


%283x-5y%29%5E2 Condense the terms.



So 18x%5E2-60xy%2B50y%5E2 completely factors to 2%283x-5y%29%5E2


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Jim