SOLUTION: Factor completely 180w^2+300wf+125f^2

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Question 541275: Factor completely
180w^2+300wf+125f^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

180w%5E2%2B300wf%2B125f%5E2 Start with the given expression.


5%2836w%5E2%2B60fw%2B25f%5E2%29 Factor out the GCF 5.


Now let's try to factor the inner expression 36w%5E2%2B60fw%2B25f%5E2


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Looking at the expression 36w%5E2%2B60fw%2B25f%5E2, we can see that the first coefficient is 36, the second coefficient is 60, and the last coefficient is 25.


Now multiply the first coefficient 36 by the last coefficient 25 to get %2836%29%2825%29=900.


Now the question is: what two whole numbers multiply to 900 (the previous product) and add to the second coefficient 60?


To find these two numbers, we need to list all of the factors of 900 (the previous product).


Factors of 900:
1,2,3,4,5,6,9,10,12,15,18,20,25,30,36,45,50,60,75,90,100,150,180,225,300,450,900
-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-25,-30,-36,-45,-50,-60,-75,-90,-100,-150,-180,-225,-300,-450,-900


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


These factors pair up and multiply to 900.
1*900 = 900
2*450 = 900
3*300 = 900
4*225 = 900
5*180 = 900
6*150 = 900
9*100 = 900
10*90 = 900
12*75 = 900
15*60 = 900
18*50 = 900
20*45 = 900
25*36 = 900
30*30 = 900
(-1)*(-900) = 900
(-2)*(-450) = 900
(-3)*(-300) = 900
(-4)*(-225) = 900
(-5)*(-180) = 900
(-6)*(-150) = 900
(-9)*(-100) = 900
(-10)*(-90) = 900
(-12)*(-75) = 900
(-15)*(-60) = 900
(-18)*(-50) = 900
(-20)*(-45) = 900
(-25)*(-36) = 900
(-30)*(-30) = 900

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 60:


First NumberSecond NumberSum
19001+900=901
24502+450=452
33003+300=303
42254+225=229
51805+180=185
61506+150=156
91009+100=109
109010+90=100
127512+75=87
156015+60=75
185018+50=68
204520+45=65
253625+36=61
303030+30=60
-1-900-1+(-900)=-901
-2-450-2+(-450)=-452
-3-300-3+(-300)=-303
-4-225-4+(-225)=-229
-5-180-5+(-180)=-185
-6-150-6+(-150)=-156
-9-100-9+(-100)=-109
-10-90-10+(-90)=-100
-12-75-12+(-75)=-87
-15-60-15+(-60)=-75
-18-50-18+(-50)=-68
-20-45-20+(-45)=-65
-25-36-25+(-36)=-61
-30-30-30+(-30)=-60



From the table, we can see that the two numbers 30 and 30 add to 60 (the middle coefficient).


So the two numbers 30 and 30 both multiply to 900 and add to 60


Now replace the middle term 60fw with 30fw%2B30fw. Remember, 30 and 30 add to 60. So this shows us that 30fw%2B30fw=60fw.


36w%5E2%2Bhighlight%2830fw%2B30fw%29%2B25f%5E2 Replace the second term 60fw with 30fw%2B30fw.


%2836w%5E2%2B30fw%29%2B%2830fw%2B25f%5E2%29 Group the terms into two pairs.


6w%286w%2B5f%29%2B%2830fw%2B25f%5E2%29 Factor out the GCF 6w from the first group.


6w%286w%2B5f%29%2B5f%286w%2B5f%29 Factor out 5f from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%286w%2B5f%29%286w%2B5f%29 Combine like terms. Or factor out the common term 6w%2B5f


%286w%2B5f%29%5E2 Condense the terms.


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So 5%2836w%5E2%2B60fw%2B25f%5E2%29 then factors further to 5%286w%2B5f%29%5E2


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Answer:


So 180w%5E2%2B300wf%2B125f%5E2 completely factors to 5%286w%2B5f%29%5E2.


In other words, 180w%5E2%2B300wf%2B125f%5E2=5%286w%2B5f%29%5E2.


Note: you can check the answer by expanding 5%286w%2B5f%29%5E2 to get 180w%5E2%2B300wf%2B125f%5E2 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

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Jim