document.write( "Question 474273: Factor Completely\r
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Algebra.Com's Answer #325285 by MathLover1(20850) $\"\"$ $\"About$

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

Looking at the expression $\"4x%5E2%2B8x-5\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"8\"$ , and the last term is $\"-5\"$ .

Now multiply the first coefficient $\"4\"$ by the last term $\"-5\"$ to get $\"%284%29%28-5%29=-20\"$ .

Now the question is: what two whole numbers multiply to $\"-20\"$ (the previous product) and add to the second coefficient $\"8\"$ ?

To find these two numbers, we need to list all of the factors of $\"-20\"$ (the previous product).

Factors of $\"-20\"$ :

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20

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

These factors pair up and multiply to $\"-20\"$ .

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"8\"$ :

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First Number	Second Number	Sum
1	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-4+5=1

From the table, we can see that the two numbers $\"-2\"$ and $\"10\"$ add to $\"8\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"10\"$ both multiply to $\"-20\"$ and add to $\"8\"$

Now replace the middle term $\"8x\"$ with $\"-2x%2B10x\"$ . Remember, $\"-2\"$ and $\"10\"$ add to $\"8\"$ . So this shows us that $\"-2x%2B10x=8x\"$ .

$\"4x%5E2%2Bhighlight%28-2x%2B10x%29-5\"$ Replace the second term $\"8x\"$ with $\"-2x%2B10x\"$ .

$\"%284x%5E2-2x%29%2B%2810x-5%29\"$ Group the terms into two pairs.

$\"2x%282x-1%29%2B%2810x-5%29\"$ Factor out the GCF $\"2x\"$ from the first group.

$\"2x%282x-1%29%2B5%282x-1%29\"$ Factor out $\"5\"$ 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.

$\"%282x%2B5%29%282x-1%29\"$ Combine like terms. Or factor out the common term $\"2x-1\"$

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Answer:

So $\"4%2Ax%5E2%2B8%2Ax-5\"$ factors to $\"%282x%2B5%29%282x-1%29\"$ .

In other words, $\"4%2Ax%5E2%2B8%2Ax-5=%282x%2B5%29%282x-1%29\"$ .

Note: you can check the answer by expanding $\"%282x%2B5%29%282x-1%29\"$ to get $\"4%2Ax%5E2%2B8%2Ax-5\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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