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

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Question 614323: factor completely: -36m^2 + 6m + 12
Answer by jim_thompson5910(33401) About Me  (Show Source):
You can put this solution on YOUR website!

-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


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:

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

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


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



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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