Question 1145014: Factor by grouping (sometimes called the ac-method).
First, choose a form with appropriate signs.
Then, fill in the blanks with numbers to be used for grouping.
Finally, show the factorization.
7x^2 + 12x - 4
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
First you are looking for two factors of the form ) and ) so that (rx\ +\ s)\ =\ prx^2\ + (ps\,+\,rq)x\ +\ qs\ =\ 7x^2\ +\ 12x\ -\ 4) , hence:
and
So, as long as we confine ourselves to the integers for the values of p, q, r, and s, we can see right away that either  and  or vice versa. For q and s, we have more choices. Clearly, one of them is negative and the other positive, so  and 
So to begin, let's start by letting
(x\ +\ s)\ =\ 7x\ +\ (7s\ +\ q)x\ +\ qs)
In order for this to be true, the following things need to be true:
Now, if  , then  which means that, in order for the sum to be a positive 12, q has to be a positive number greater than 12. Since the only positive choices we have for  are 1, 2, and 4, is not possible and we know that  or  or 
If , then  and so \ -\ 1\ =\ 27\ \neq\ 12)
If , then  and so \ -\ 4\ =\ 3\ \neq\ 12)
If , then  and so \ -\ 2\ =\ 12)
Therefore
(x\ +\ 2))
John
My calculator said it, I believe it, that settles it
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