document.write( "Question 463845: Factor completely, if it is prime state this. -12x^2 – 28x + 24 \n" ); document.write( "

Algebra.Com's Answer #317724 by MathLover1(20850) $\"\"$ $\"About$

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

$\"-12%2Ax%5E2-28%2Ax%2B24\"$ Start with the given expression.

$\"-4%283x%5E2%2B7x-6%29\"$ Factor out the GCF $\"-4\"$ .

Now let's try to factor the inner expression $\"3x%5E2%2B7x-6\"$

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Looking at the expression $\"3x%5E2%2B7x-6\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-6\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-6\"$ to get $\"%283%29%28-6%29=-18\"$ .

Now the question is: what two whole numbers multiply to $\"-18\"$ (the previous product) and add to the second coefficient $\"7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-18\"$ (the previous product).

Factors of $\"-18\"$ :

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18

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

These factors pair up and multiply to $\"-18\"$ .

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"7\"$ :

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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

From the table, we can see that the two numbers $\"-2\"$ and $\"9\"$ add to $\"7\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"9\"$ both multiply to $\"-18\"$ and add to $\"7\"$

Now replace the middle term $\"7x\"$ with $\"-2x%2B9x\"$ . Remember, $\"-2\"$ and $\"9\"$ add to $\"7\"$ . So this shows us that $\"-2x%2B9x=7x\"$ .

$\"3x%5E2%2Bhighlight%28-2x%2B9x%29-6\"$ Replace the second term $\"7x\"$ with $\"-2x%2B9x\"$ .

$\"%283x%5E2-2x%29%2B%289x-6%29\"$ Group the terms into two pairs.

$\"x%283x-2%29%2B%289x-6%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%283x-2%29%2B3%283x-2%29\"$ Factor out $\"3\"$ 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.

$\"%28x%2B3%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$

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So $\"-4%283x%5E2%2B7x-6%29\"$ then factors further to $\"-4%28x%2B3%29%283x-2%29\"$

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Answer:

So $\"-12%2Ax%5E2-28%2Ax%2B24\"$ completely factors to $\"-4%28x%2B3%29%283x-2%29\"$ .

In other words, $\"-12%2Ax%5E2-28%2Ax%2B24=-4%28x%2B3%29%283x-2%29\"$ .

Note: you can check the answer by expanding $\"-4%28x%2B3%29%283x-2%29\"$ to get $\"-12%2Ax%5E2-28%2Ax%2B24\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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