SOLUTION: 24x+2x-12

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Question 127884: 24x+2x-12
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Do you mean 24x%5E2%2B2x-12? Do you want to factor?




24x%5E2%2B2x-12 Start with the given expression


2%2812x%5E2%2Bx-6%29 Factor out the GCF 2


Now let's focus on the inner expression 12x%5E2%2Bx-6




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Looking at 12x%5E2%2B1x-6 we can see that the first term is 12x%5E2 and the last term is -6 where the coefficients are 12 and -6 respectively.

Now multiply the first coefficient 12 and the last coefficient -6 to get -72. Now what two numbers multiply to -72 and add to the middle coefficient 1? Let's list all of the factors of -72:



Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72

-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -72
(1)*(-72)
(2)*(-36)
(3)*(-24)
(4)*(-18)
(6)*(-12)
(8)*(-9)
(-1)*(72)
(-2)*(36)
(-3)*(24)
(-4)*(18)
(-6)*(12)
(-8)*(9)

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
1-721+(-72)=-71
2-362+(-36)=-34
3-243+(-24)=-21
4-184+(-18)=-14
6-126+(-12)=-6
8-98+(-9)=-1
-172-1+72=71
-236-2+36=34
-324-3+24=21
-418-4+18=14
-612-6+12=6
-89-8+9=1



From this list we can see that -8 and 9 add up to 1 and multiply to -72


Now looking at the expression 12x%5E2%2B1x-6, replace 1x with -8x%2B9x (notice -8x%2B9x adds up to 1x. So it is equivalent to 1x)

12x%5E2%2Bhighlight%28-8x%2B9x%29%2B-6


Now let's factor 12x%5E2-8x%2B9x-6 by grouping:


%2812x%5E2-8x%29%2B%289x-6%29 Group like terms


4x%283x-2%29%2B3%283x-2%29 Factor out the GCF of 4x out of the first group. Factor out the GCF of 3 out of the second group


%284x%2B3%29%283x-2%29 Since we have a common term of 3x-2, we can combine like terms

So 12x%5E2-8x%2B9x-6 factors to %284x%2B3%29%283x-2%29


So this also means that 12x%5E2%2B1x-6 factors to %284x%2B3%29%283x-2%29 (since 12x%5E2%2B1x-6 is equivalent to 12x%5E2-8x%2B9x-6)



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So our expression goes from 2%2812x%5E2%2Bx-6%29 and factors further to 2%284x%2B3%29%283x-2%29


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Answer:

So 24x%5E2%2B2x-12 factors to 2%284x%2B3%29%283x-2%29