SOLUTION: Factor by grouping 21a^2-28ab-28b^2

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Question 191754: Factor by grouping
21a^2-28ab-28b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

21a%5E2-28ab-28b%5E2 Start with the given expression


7%283a%5E2-4ab-4b%5E2%29 Factor out the GCF 7


Now let's focus on the inner expression 3a%5E2-4ab-4b%5E2




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Looking at 3a%5E2-4ab-4b%5E2 we can see that the first term is 3a%5E2 and the last term is -4b%5E2 where the coefficients are 3 and -4 respectively.

Now multiply the first coefficient 3 and the last coefficient -4 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient -4? Let's list all of the factors of -12:



Factors of -12:
1,2,3,4,6,12

-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -12
(1)*(-12)
(2)*(-6)
(3)*(-4)
(-1)*(12)
(-2)*(6)
(-3)*(4)

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4

First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1



From this list we can see that 2 and -6 add up to -4 and multiply to -12


Now looking at the expression 3a%5E2-4ab-4b%5E2, replace -4ab with 2ab-6ab (notice 2ab-6ab adds up to -4ab. So it is equivalent to -4ab)

3a%5E2%2Bhighlight%282ab-6ab%29-4b%5E2


Now let's factor 3a%5E2%2B2ab-6ab-4b%5E2 by grouping:


%283a%5E2%2B2ab%29%2B%28-6ab-4b%5E2%29 Group like terms


a%283a%2B2b%29-2b%283a%2B2b%29 Factor out the GCF of a out of the first group. Factor out the GCF of -2b out of the second group


%28a-2b%29%283a%2B2b%29 Since we have a common term of 3a%2B2b, we can combine like terms

So 3a%5E2%2B2ab-6ab-4b%5E2 factors to %28a-2b%29%283a%2B2b%29


So this also means that 3a%5E2-4ab-4b%5E2 factors to %28a-2b%29%283a%2B2b%29 (since 3a%5E2-4ab-4b%5E2 is equivalent to 3a%5E2%2B2ab-6ab-4b%5E2)



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So our expression goes from 7%283a%5E2-4ab-4b%5E2%29 and factors further to 7%28a-2b%29%283a%2B2b%29


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Answer:

So 21a%5E2-28ab-28b%5E2 completely factors to 7%28a-2b%29%283a%2B2b%29