SOLUTION: Factoring 6x^2 -28x -48

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Question 99714: Factoring 6x^2 -28x -48
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


6%2Ax%5E2-28%2Ax-48 Start with the given expression.



2%283x%5E2-14x-24%29 Factor out the GCF 2.



Now let's try to factor the inner expression 3x%5E2-14x-24



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Looking at the expression 3x%5E2-14x-24, we can see that the first coefficient is 3, the second coefficient is -14, and the last term is -24.



Now multiply the first coefficient 3 by the last term -24 to get %283%29%28-24%29=-72.



Now the question is: what two whole numbers multiply to -72 (the previous product) and add to the second coefficient -14?



To find these two numbers, we need to list all of the factors of -72 (the previous product).



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



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



These factors pair up and multiply to -72.

1*(-72) = -72
2*(-36) = -72
3*(-24) = -72
4*(-18) = -72
6*(-12) = -72
8*(-9) = -72
(-1)*(72) = -72
(-2)*(36) = -72
(-3)*(24) = -72
(-4)*(18) = -72
(-6)*(12) = -72
(-8)*(9) = -72


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -14:



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 the table, we can see that the two numbers 4 and -18 add to -14 (the middle coefficient).



So the two numbers 4 and -18 both multiply to -72 and add to -14



Now replace the middle term -14x with 4x-18x. Remember, 4 and -18 add to -14. So this shows us that 4x-18x=-14x.



3x%5E2%2Bhighlight%284x-18x%29-24 Replace the second term -14x with 4x-18x.



%283x%5E2%2B4x%29%2B%28-18x-24%29 Group the terms into two pairs.



x%283x%2B4%29%2B%28-18x-24%29 Factor out the GCF x from the first group.



x%283x%2B4%29-6%283x%2B4%29 Factor out 6 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.



%28x-6%29%283x%2B4%29 Combine like terms. Or factor out the common term 3x%2B4



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So 2%283x%5E2-14x-24%29 then factors further to 2%28x-6%29%283x%2B4%29



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Answer:



So 6%2Ax%5E2-28%2Ax-48 completely factors to 2%28x-6%29%283x%2B4%29.



In other words, 6%2Ax%5E2-28%2Ax-48=2%28x-6%29%283x%2B4%29.



Note: you can check the answer by expanding 2%28x-6%29%283x%2B4%29 to get 6%2Ax%5E2-28%2Ax-48 or by graphing the original expression and the answer (the two graphs should be identical).