SOLUTION: factor 32a^2+4a-6

Question 119790: factor 32a^2+4a-6
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given expression

Factor out the GCF

Now let's focus on the inner expression

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Looking at we can see that the first term is and the last term is where the coefficients are 16 and -3 respectively.

Now multiply the first coefficient 16 and the last coefficient -3 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient 2? Let's list all of the factors of -48:

Factors of -48:
1,2,3,4,6,8,12,16,24,48

-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -48
(1)*(-48)
(2)*(-24)
(3)*(-16)
(4)*(-12)
(6)*(-8)
(-1)*(48)
(-2)*(24)
(-3)*(16)
(-4)*(12)
(-6)*(8)

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

Now which of these pairs add to 2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 2

First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

From this list we can see that -6 and 8 add up to 2 and multiply to -48

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

Now let's factor by grouping:

Group like terms

Factor out the GCF of out of the first group. Factor out the GCF of out of the second group

Since we have a common term of , we can combine like terms

So factors to

So this also means that factors to (since is equivalent to )

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Now reintroduce the GCF

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Answer:

So factors to

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