document.write( "Question 590972: 24x2-42x-12=0 \n" ); document.write( "

Algebra.Com's Answer #375417 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

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

$\"24%2Ax%5E2-42%2Ax-12\"$ Start with the given expression.

$\"6%284x%5E2-7x-2%29\"$ Factor out the GCF $\"6\"$ .

Now let's try to factor the inner expression $\"4x%5E2-7x-2\"$

---------------------------------------------------------------

Looking at the expression $\"4x%5E2-7x-2\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-7\"$ , and the last term is $\"-2\"$ .

Now multiply the first coefficient $\"4\"$ by the last term $\"-2\"$ to get $\"%284%29%28-2%29=-8\"$ .

Now the question is: what two whole numbers multiply to $\"-8\"$ (the previous product) and add to the second coefficient $\"-7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-8\"$ (the previous product).

Factors of $\"-8\"$ :

1,2,4,8

-1,-2,-4,-8

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

These factors pair up and multiply to $\"-8\"$ .

1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8

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

\n" ); document.write( "

First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-2+4=2

From the table, we can see that the two numbers $\"1\"$ and $\"-8\"$ add to $\"-7\"$ (the middle coefficient).

So the two numbers $\"1\"$ and $\"-8\"$ both multiply to $\"-8\"$ and add to $\"-7\"$

Now replace the middle term $\"-7x\"$ with $\"x-8x\"$ . Remember, $\"1\"$ and $\"-8\"$ add to $\"-7\"$ . So this shows us that $\"x-8x=-7x\"$ .

$\"4x%5E2%2Bhighlight%28x-8x%29-2\"$ Replace the second term $\"-7x\"$ with $\"x-8x\"$ .

$\"%284x%5E2%2Bx%29%2B%28-8x-2%29\"$ Group the terms into two pairs.

$\"x%284x%2B1%29%2B%28-8x-2%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%284x%2B1%29-2%284x%2B1%29\"$ 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.

$\"%28x-2%29%284x%2B1%29\"$ Combine like terms. Or factor out the common term $\"4x%2B1\"$

--------------------------------------------------

So $\"6%284x%5E2-7x-2%29\"$ then factors further to $\"6%28x-2%29%284x%2B1%29\"$

===============================================================

Answer:

So $\"24%2Ax%5E2-42%2Ax-12\"$ completely factors to $\"6%28x-2%29%284x%2B1%29\"$ .

In other words, $\"24%2Ax%5E2-42%2Ax-12=6%28x-2%29%284x%2B1%29\"$ .

Note: you can check the answer by expanding $\"6%28x-2%29%284x%2B1%29\"$ to get $\"24%2Ax%5E2-42%2Ax-12\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( " \n" ); document.write( "

\n" );