document.write( "Question 270245: how can i factor this completely?\r
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Algebra.Com's Answer #197999 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"2x%5E2%2B7x-15\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-15\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"-15\"$ to get $\"%282%29%28-15%29=-30\"$ .

Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-30\"$ (the previous product).

Factors of $\"-30\"$ :

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30

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

These factors pair up and multiply to $\"-30\"$ .

1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30

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

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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

From the table, we can see that the two numbers $\"-3\"$ and $\"10\"$ add to $\"7\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"10\"$ both multiply to $\"-30\"$ and add to $\"7\"$

Now replace the middle term $\"7x\"$ with $\"-3x%2B10x\"$ . Remember, $\"-3\"$ and $\"10\"$ add to $\"7\"$ . So this shows us that $\"-3x%2B10x=7x\"$ .

$\"2x%5E2%2Bhighlight%28-3x%2B10x%29-15\"$ Replace the second term $\"7x\"$ with $\"-3x%2B10x\"$ .

$\"%282x%5E2-3x%29%2B%2810x-15%29\"$ Group the terms into two pairs.

$\"x%282x-3%29%2B%2810x-15%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%282x-3%29%2B5%282x-3%29\"$ 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.

$\"%28x%2B5%29%282x-3%29\"$ Combine like terms. Or factor out the common term $\"2x-3\"$

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Answer:

So $\"2%2Ax%5E2%2B7%2Ax-15\"$ factors to $\"%28x%2B5%29%282x-3%29\"$ .

In other words, $\"2%2Ax%5E2%2B7%2Ax-15=%28x%2B5%29%282x-3%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B5%29%282x-3%29\"$ to get $\"2%2Ax%5E2%2B7%2Ax-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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