document.write( "Question 283992: how to factor completely: 16x^2 + 4x - 2 = ? \n" ); document.write( "

Algebra.Com's Answer #206095 by richwmiller(17219) $\"\"$ $\"About$

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16x^2 + 4x - 2
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"8x%5E2%2B2x-1\"$ , we can see that the first coefficient is $\"8\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-1\"$ .

Now multiply the first coefficient $\"8\"$ by the last term $\"-1\"$ to get $\"%288%29%28-1%29=-8\"$ .

Now the question is: what two whole numbers multiply to $\"-8\"$ (the previous product) and add to the second coefficient $\"2\"$ ?

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 $\"2\"$ :

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

So the two numbers $\"-2\"$ and $\"4\"$ both multiply to $\"-8\"$ and add to $\"2\"$

Now replace the middle term $\"2x\"$ with $\"-2x%2B4x\"$ . Remember, $\"-2\"$ and $\"4\"$ add to $\"2\"$ . So this shows us that $\"-2x%2B4x=2x\"$ .

$\"8x%5E2%2Bhighlight%28-2x%2B4x%29-1\"$ Replace the second term $\"2x\"$ with $\"-2x%2B4x\"$ .

$\"%288x%5E2-2x%29%2B%284x-1%29\"$ Group the terms into two pairs.

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

$\"2x%284x-1%29%2B1%284x-1%29\"$ Factor out $\"1\"$ 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.

$\"%282x%2B1%29%284x-1%29\"$ Combine like terms. Or factor out the common term $\"4x-1\"$

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Answer:

So $\"8%2Ax%5E2%2B2%2Ax-1\"$ factors to $\"%282x%2B1%29%284x-1%29\"$ .

In other words, $\"8%2Ax%5E2%2B2%2Ax-1=%282x%2B1%29%284x-1%29\"$ .

Note: you can check the answer by expanding $\"%282x%2B1%29%284x-1%29\"$ to get $\"8%2Ax%5E2%2B2%2Ax-1\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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