document.write( "Question 276480: Please help me answer this question.
\n" ); document.write( "What are the linear factors of 12x^2+7x-12?
\n" ); document.write( "The answer given is 4x-3 and 3x+4.
\n" ); document.write( "I'm just not too sure how to get to this answer. \n" ); document.write( "

Algebra.Com's Answer #201553 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"12x%5E2%2B7x-12\"$ , we can see that the first coefficient is $\"12\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-12\"$ .

Now multiply the first coefficient $\"12\"$ by the last term $\"-12\"$ to get $\"%2812%29%28-12%29=-144\"$ .

Now the question is: what two whole numbers multiply to $\"-144\"$ (the previous product) and add to the second coefficient $\"7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-144\"$ (the previous product).

Factors of $\"-144\"$ :

1,2,3,4,6,8,9,12,16,18,24,36,48,72,144

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144

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

These factors pair up and multiply to $\"-144\"$ .

1*(-144) = -144
2*(-72) = -144
3*(-48) = -144
4*(-36) = -144
6*(-24) = -144
8*(-18) = -144
9*(-16) = -144
12*(-12) = -144
(-1)*(144) = -144
(-2)*(72) = -144
(-3)*(48) = -144
(-4)*(36) = -144
(-6)*(24) = -144
(-8)*(18) = -144
(-9)*(16) = -144
(-12)*(12) = -144

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	-144	1+(-144)=-143
2	-72	2+(-72)=-70
3	-48	3+(-48)=-45
4	-36	4+(-36)=-32
6	-24	6+(-24)=-18
8	-18	8+(-18)=-10
9	-16	9+(-16)=-7
12	-12	12+(-12)=0
-1	144	-1+144=143
-2	72	-2+72=70
-3	48	-3+48=45
-4	36	-4+36=32
-6	24	-6+24=18
-8	18	-8+18=10
-9	16	-9+16=7
-12	12	-12+12=0

From the table, we can see that the two numbers $\"-9\"$ and $\"16\"$ add to $\"7\"$ (the middle coefficient).

So the two numbers $\"-9\"$ and $\"16\"$ both multiply to $\"-144\"$ and add to $\"7\"$

Now replace the middle term $\"7x\"$ with $\"-9x%2B16x\"$ . Remember, $\"-9\"$ and $\"16\"$ add to $\"7\"$ . So this shows us that $\"-9x%2B16x=7x\"$ .

$\"12x%5E2%2Bhighlight%28-9x%2B16x%29-12\"$ Replace the second term $\"7x\"$ with $\"-9x%2B16x\"$ .

$\"%2812x%5E2-9x%29%2B%2816x-12%29\"$ Group the terms into two pairs.

$\"3x%284x-3%29%2B%2816x-12%29\"$ Factor out the GCF $\"3x\"$ from the first group.

$\"3x%284x-3%29%2B4%284x-3%29\"$ Factor out $\"4\"$ 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.

$\"%283x%2B4%29%284x-3%29\"$ Combine like terms. Or factor out the common term $\"4x-3\"$

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Answer:

So $\"12%2Ax%5E2%2B7%2Ax-12\"$ factors to $\"%283x%2B4%29%284x-3%29\"$ .

In other words, $\"12%2Ax%5E2%2B7%2Ax-12=%283x%2B4%29%284x-3%29\"$ .

Note: you can check the answer by expanding $\"%283x%2B4%29%284x-3%29\"$ to get $\"12%2Ax%5E2%2B7%2Ax-12\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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