SOLUTION: Factor completely as possible. 9x^2-39x-30 Factor by grouping 10x^2-4xy-15xy+6y^2

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Question 164344: Factor completely as possible.
9x^2-39x-30
Factor by grouping
10x^2-4xy-15xy+6y^2
Found 2 solutions by jim_thompson5910, Earlsdon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'll do the first one to get you started

# 1

9x%5E2-39x-30 Start with the given expression


3%283x%5E2-13x-10%29 Factor out the GCF 3


Now let's focus on the inner expression 3x%5E2-13x-10


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Looking at the expression 3x%5E2-13x-10, we can see that the first coefficient is 3, the second coefficient is -13, and the last term is -10.


Now multiply the first coefficient 3 by the last term -10 to get %283%29%28-10%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient -13?


To find these two numbers, we need to list all of the factors of -30 (the previous product).


Factors of -30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to -30.
1*(-30)
2*(-15)
3*(-10)
5*(-6)
(-1)*(30)
(-2)*(15)
(-3)*(10)
(-5)*(6)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -13:


First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From the table, we can see that the two numbers 2 and -15 add to -13 (the middle coefficient).


So the two numbers 2 and -15 both multiply to -30 and add to -13


Now replace the middle term -13x with 2x-15x. Remember, 2 and -15 add to -13. So this shows us that 2x-15x=-13x.


3x%5E2%2Bhighlight%282x-15x%29-10 Replace the second term -13x with 2x-15x.


%283x%5E2%2B2x%29%2B%28-15x-10%29 Group the terms into two pairs.


x%283x%2B2%29%2B%28-15x-10%29 Factor out the GCF x from the first group.


x%283x%2B2%29-5%283x%2B2%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.


%28x-5%29%283x%2B2%29 Combine like terms. Or factor out the common term 3x%2B2


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So our expression goes from 3%283x%5E2-13x-10%29 and factors further to 3%28x-5%29%283x%2B2%29


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Answer:

So 9x%5E2-39x-30 completely factors to 3%28x-5%29%283x%2B2%29

Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
Factor:
9x%5E2-39x-30 First, notice that 3 is a common factor of each of the three terms, so you can factor this.
3%283x%5E2-13x-10%29 Now factor the trinomial in parentheses.
highlight%283%283x%2B2%29%28x-5%29%29
Factor by grouping:
10x%5E2-4xy-15xy%2B6y%5E2 Group the terms as shown below:
%2810x%5E2-4xy%29-%2815xy-6y%5E2%29 Notice the change of sign on the last term! Now look for common factors in the terms of each group.
2x%285x-2y%29-3y%285x-2y%29 Now you can factor the common factors of (5x-2y)
highlight%28%285x-2y%29%282x-3y%29%29