document.write( "Question 284203: factor
\n" ); document.write( "18x^3-60x^2+50x \n" ); document.write( "

Algebra.Com's Answer #206209 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
first thing is factor out 2x
\n" ); document.write( "2x*(9x^-30x+25)
\n" ); document.write( "then 9*25=225
\n" ); document.write( "so we want factors of 225 that add to 30
\n" ); document.write( "15^2=225
\n" ); document.write( "(3x-5)^2
\n" ); document.write( "finally 2x*(3x-5)^2
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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 $\"-30\"$ :

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First Number	Second Number	Sum
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
-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.

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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).

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