document.write( "Question 259609: One of the solutions of the equation: 3x^2-2x-8=0 \n" ); document.write( "

Algebra.Com's Answer #191088 by richwmiller(17219) $\"\"$ $\"About$

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

Looking at the expression $\"3x%5E2-2x-8\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"-8\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-8\"$ to get $\"%283%29%28-8%29=-24\"$ .

Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?

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 $\"-2\"$ :

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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 $\"4\"$ and $\"-6\"$ add to $\"-2\"$ (the middle coefficient).

So the two numbers $\"4\"$ and $\"-6\"$ both multiply to $\"-24\"$ and add to $\"-2\"$

Now replace the middle term $\"-2x\"$ with $\"4x-6x\"$ . Remember, $\"4\"$ and $\"-6\"$ add to $\"-2\"$ . So this shows us that $\"4x-6x=-2x\"$ .

$\"3x%5E2%2Bhighlight%284x-6x%29-8\"$ Replace the second term $\"-2x\"$ with $\"4x-6x\"$ .

$\"%283x%5E2%2B4x%29%2B%28-6x-8%29\"$ Group the terms into two pairs.

$\"x%283x%2B4%29%2B%28-6x-8%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%283x%2B4%29-2%283x%2B4%29\"$ 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.

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

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Answer:

So $\"3%2Ax%5E2-2%2Ax-8\"$ factors to $\"%28x-2%29%283x%2B4%29\"$ .

In other words, $\"3%2Ax%5E2-2%2Ax-8=%28x-2%29%283x%2B4%29\"$ .

Note: you can check the answer by expanding $\"%28x-2%29%283x%2B4%29\"$ to get $\"3%2Ax%5E2-2%2Ax-8\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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