SOLUTION: Factor completely 6a^2 – 6a - 72

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Question 189830: Factor completely
6a^2 – 6a - 72

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
6a%5E2-6a-72 Start with the given expression


6%28a%5E2-a-12%29 Factor out the GCF 6


Now let's focus on the inner expression a%5E2-a-12




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Looking at a%5E2-a-12 we can see that the first term is a%5E2 and the last term is -12 where the coefficients are 1 and -12 respectively.

Now multiply the first coefficient 1 and the last coefficient -12 to get -12. Now what two numbers multiply to -12 and add to the middle coefficient -1? 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 -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1

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 3 and -4 add up to -1 and multiply to -12


Now looking at the expression a%5E2-a-12, replace -a with 3a-4a (notice 3a-4a combines to -a. So it is equivalent to -a)

a%5E2%2Bhighlight%283a-4a%29-12


Now let's factor a%5E2%2B3a-4a-12 by grouping:


%28a%5E2%2B3a%29%2B%28-4a-12%29 Group like terms


a%28a%2B3%29-4%28a%2B3%29 Factor out the GCF of a out of the first group. Factor out the GCF of -4 out of the second group


%28a-4%29%28a%2B3%29 Since we have a common term of a%2B3, we can combine like terms

So a%5E2%2B3a-4a-12 factors to %28a-4%29%28a%2B3%29


So this also means that a%5E2-a-12 factors to %28a-4%29%28a%2B3%29 (since a%5E2-a-12 is equivalent to a%5E2%2B3a-4a-12)



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So 6%28a%5E2-a-12%29 factors to 6%28a-4%29%28a%2B3%29


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Answer:


So 6a%5E2-6a-72 completely factors to 6%28a-4%29%28a%2B3%29