document.write( "Question 111424: Factor completely:
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Algebra.Com's Answer #81255 by jim_thompson5910(35256) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"24%2Ax%5E2%2B10%2Ax-4\"$ Start with the given expression.

$\"2%2812x%5E2%2B5x-2%29\"$ Factor out the GCF $\"2\"$ .

Now let's try to factor the inner expression $\"12x%5E2%2B5x-2\"$

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Looking at the expression $\"12x%5E2%2B5x-2\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-2\"$ .

Now multiply the first coefficient $\"12\"$ by the last term $\"-2\"$ to get $\"%2812%29%28-2%29=-24\"$ .

Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"5\"$ ?

To find these two numbers, we need to list all of the factors of $\"-24\"$ (the previous product).

Factors of $\"-24\"$ :

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24

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

These factors pair up and multiply to $\"-24\"$ .

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"5\"$ :

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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

From the table, we can see that the two numbers $\"-3\"$ and $\"8\"$ add to $\"5\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"8\"$ both multiply to $\"-24\"$ and add to $\"5\"$

Now replace the middle term $\"5x\"$ with $\"-3x%2B8x\"$ . Remember, $\"-3\"$ and $\"8\"$ add to $\"5\"$ . So this shows us that $\"-3x%2B8x=5x\"$ .

$\"12x%5E2%2Bhighlight%28-3x%2B8x%29-2\"$ Replace the second term $\"5x\"$ with $\"-3x%2B8x\"$ .

$\"%2812x%5E2-3x%29%2B%288x-2%29\"$ Group the terms into two pairs.

$\"3x%284x-1%29%2B%288x-2%29\"$ Factor out the GCF $\"3x\"$ from the first group.

$\"3x%284x-1%29%2B2%284x-1%29\"$ Factor out $\"2\"$ 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.

$\"%283x%2B2%29%284x-1%29\"$ Combine like terms. Or factor out the common term $\"4x-1\"$

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So $\"2%2812x%5E2%2B5x-2%29\"$ then factors further to $\"2%283x%2B2%29%284x-1%29\"$

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Answer:

So $\"24%2Ax%5E2%2B10%2Ax-4\"$ completely factors to $\"2%283x%2B2%29%284x-1%29\"$ .

In other words, $\"24%2Ax%5E2%2B10%2Ax-4=2%283x%2B2%29%284x-1%29\"$ .

Note: you can check the answer by expanding $\"2%283x%2B2%29%284x-1%29\"$ to get $\"24%2Ax%5E2%2B10%2Ax-4\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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