SOLUTION: 75b^2-180b+108

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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