SOLUTION: factor 18x^3-60x^2+50x

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Question 284203: factor
18x^3-60x^2+50x

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
first thing is factor out 2x
2x*(9x^-30x+25)
then 9*25=225
so we want factors of 225 that add to 30
15^2=225
(3x-5)^2
finally 2x*(3x-5)^2
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 9x%5E2-30x%2B25, we can see that the first coefficient is 9, the second coefficient is -30, and the last term is 25.



Now multiply the first coefficient 9 by the last term 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 -30x with -15x-15x. Remember, -15 and -15 add to -30. So this shows us that -15x-15x=-30x.



9x%5E2%2Bhighlight%28-15x-15x%29%2B25 Replace the second term -30x with -15x-15x.



%289x%5E2-15x%29%2B%28-15x%2B25%29 Group the terms into two pairs.



3x%283x-5%29%2B%28-15x%2B25%29 Factor out the GCF 3x from the first group.



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



%283x-5%29%283x-5%29 Combine like terms. Or factor out the common term 3x-5



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



===============================================================



Answer:



So 9%2Ax%5E2-30%2Ax%2B25 factors to %283x-5%29%5E2.



In other words, 9%2Ax%5E2-30%2Ax%2B25=%283x-5%29%5E2.



Note: you can check the answer by expanding %283x-5%29%5E2 to get 9%2Ax%5E2-30%2Ax%2B25 or by graphing the original expression and the answer (the two graphs should be identical).