SOLUTION: factor completely: -36m^2 + 6m + 12

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Question 614323: factor completely: -36m^2 + 6m + 12
Answer by jim_thompson5910(35256) About Me  (Show Source):
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-36m%5E2%2B6m%2B12 Start with the given expression.


-6%286m%5E2-m-2%29 Factor out the GCF -6.


Now let's try to factor the inner expression 6m%5E2-m-2


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Looking at the expression 6m%5E2-m-2, we can see that the first coefficient is 6, the second coefficient is -1, and the last term is -2.


Now multiply the first coefficient 6 by the last term -2 to get %286%29%28-2%29=-12.


Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient -1?


To find these two numbers, we need to list all of the factors of -12 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -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 the table, we can see that the two numbers 3 and -4 add to -1 (the middle coefficient).


So the two numbers 3 and -4 both multiply to -12 and add to -1


Now replace the middle term -1m with 3m-4m. Remember, 3 and -4 add to -1. So this shows us that 3m-4m=-1m.


6m%5E2%2Bhighlight%283m-4m%29-2 Replace the second term -1m with 3m-4m.


%286m%5E2%2B3m%29%2B%28-4m-2%29 Group the terms into two pairs.


3m%282m%2B1%29%2B%28-4m-2%29 Factor out the GCF 3m from the first group.


3m%282m%2B1%29-2%282m%2B1%29 Factor out 2 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.


%283m-2%29%282m%2B1%29 Combine like terms. Or factor out the common term 2m%2B1


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So -6%286m%5E2-m-2%29 then factors further to -6%283m-2%29%282m%2B1%29


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Answer:


So -36m%5E2%2B6m%2B12 completely factors to -6%283m-2%29%282m%2B1%29.


In other words, -36m%5E2%2B6m%2B12=-6%283m-2%29%282m%2B1%29.


Note: you can check the answer by expanding -6%283m-2%29%282m%2B1%29 to get -36m%5E2%2B6m%2B12 or by graphing the original expression and the answer (the two graphs should be identical).

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