document.write( "Question 91478: Factor\r
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Algebra.Com's Answer #66425 by jim_thompson5910(35256) $\"\"$ $\"About$

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

Looking at the expression $\"2x%5E2%2B5x-25\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-25\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"-25\"$ to get $\"%282%29%28-25%29=-50\"$ .

Now the question is: what two whole numbers multiply to $\"-50\"$ (the previous product) and add to the second coefficient $\"5\"$ ?

To find these two numbers, we need to list all of the factors of $\"-50\"$ (the previous product).

Factors of $\"-50\"$ :

1,2,5,10,25,50

-1,-2,-5,-10,-25,-50

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

These factors pair up and multiply to $\"-50\"$ .

1*(-50) = -50
2*(-25) = -50
5*(-10) = -50
(-1)*(50) = -50
(-2)*(25) = -50
(-5)*(10) = -50

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"5\"$ :

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First Number	Second Number	Sum
1	-50	1+(-50)=-49
2	-25	2+(-25)=-23
5	-10	5+(-10)=-5
-1	50	-1+50=49
-2	25	-2+25=23
-5	10	-5+10=5

From the table, we can see that the two numbers $\"-5\"$ and $\"10\"$ add to $\"5\"$ (the middle coefficient).

So the two numbers $\"-5\"$ and $\"10\"$ both multiply to $\"-50\"$ and add to $\"5\"$

Now replace the middle term $\"5x\"$ with $\"-5x%2B10x\"$ . Remember, $\"-5\"$ and $\"10\"$ add to $\"5\"$ . So this shows us that $\"-5x%2B10x=5x\"$ .

$\"2x%5E2%2Bhighlight%28-5x%2B10x%29-25\"$ Replace the second term $\"5x\"$ with $\"-5x%2B10x\"$ .

$\"%282x%5E2-5x%29%2B%2810x-25%29\"$ Group the terms into two pairs.

$\"x%282x-5%29%2B%2810x-25%29\"$ Factor out the GCF $\"x\"$ from the first group.

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

$\"%28x%2B5%29%282x-5%29\"$ Combine like terms. Or factor out the common term $\"2x-5\"$

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Answer:

So $\"2%2Ax%5E2%2B5%2Ax-25\"$ factors to $\"%28x%2B5%29%282x-5%29\"$ .

In other words, $\"2%2Ax%5E2%2B5%2Ax-25=%28x%2B5%29%282x-5%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B5%29%282x-5%29\"$ to get $\"2%2Ax%5E2%2B5%2Ax-25\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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