document.write( "Question 99714: Factoring 6x^2 -28x -48 \n" ); document.write( "

Algebra.Com's Answer #72585 by jim_thompson5910(35256) $\"\"$ $\"About$

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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 $\"-14\"$ :

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First Number	Second Number	Sum
1	-72	1+(-72)=-71
2	-36	2+(-36)=-34
3	-24	3+(-24)=-21
4	-18	4+(-18)=-14
6	-12	6+(-12)=-6
8	-9	8+(-9)=-1
-1	72	-1+72=71
-2	36	-2+36=34
-3	24	-3+24=21
-4	18	-4+18=14
-6	12	-6+12=6
-8	9	-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).

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