Question 171351: 75b^2-180b+108
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! I'm assuming that you want to factor
 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 25 and 36 respectively.
Now multiply the first coefficient 25 and the last coefficient 36 to get 900. Now what two numbers multiply to 900 and add to the middle coefficient -60? Let's list all of the factors of 900:
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
-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 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 900
1*900
2*450
3*300
4*225
5*180
6*150
9*100
10*90
12*75
15*60
18*50
20*45
25*36
30*30
(-1)*(-900)
(-2)*(-450)
(-3)*(-300)
(-4)*(-225)
(-5)*(-180)
(-6)*(-150)
(-9)*(-100)
(-10)*(-90)
(-12)*(-75)
(-15)*(-60)
(-18)*(-50)
(-20)*(-45)
(-25)*(-36)
(-30)*(-30)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -60? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -60
| First Number | Second Number | Sum | | 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 | | -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 this list we can see that -30 and -30 add up to -60 and multiply to 900
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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