document.write( "Question 824618: Solve the equation
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Algebra.Com's Answer #496615 by richwmiller(17219) $\"\"$ $\"About$

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2x^2-11x-21 = 4x-28
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"2x%5E2-15x%2B7\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-15\"$ , and the last term is $\"7\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"7\"$ to get $\"%282%29%287%29=14\"$ .

Now the question is: what two whole numbers multiply to $\"14\"$ (the previous product) and add to the second coefficient $\"-15\"$ ?

To find these two numbers, we need to list all of the factors of $\"14\"$ (the previous product).

Factors of $\"14\"$ :

1,2,7,14

-1,-2,-7,-14

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

These factors pair up and multiply to $\"14\"$ .

1*14 = 14
2*7 = 14
(-1)*(-14) = 14
(-2)*(-7) = 14

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-15\"$ :

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First Number	Second Number	Sum
1	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

From the table, we can see that the two numbers $\"-1\"$ and $\"-14\"$ add to $\"-15\"$ (the middle coefficient).

So the two numbers $\"-1\"$ and $\"-14\"$ both multiply to $\"14\"$ and add to $\"-15\"$

Now replace the middle term $\"-15x\"$ with $\"-x-14x\"$ . Remember, $\"-1\"$ and $\"-14\"$ add to $\"-15\"$ . So this shows us that $\"-x-14x=-15x\"$ .

$\"2x%5E2%2Bhighlight%28-x-14x%29%2B7\"$ Replace the second term $\"-15x\"$ with $\"-x-14x\"$ .

$\"%282x%5E2-x%29%2B%28-14x%2B7%29\"$ Group the terms into two pairs.

$\"x%282x-1%29%2B%28-14x%2B7%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%282x-1%29-7%282x-1%29\"$ Factor out $\"7\"$ 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.

$\"%28x-7%29%282x-1%29\"$ Combine like terms. Or factor out the common term $\"2x-1\"$

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Answer:

So $\"2%2Ax%5E2-15%2Ax%2B7\"$ factors to $\"%28x-7%29%282x-1%29\"$ .

In other words, $\"2%2Ax%5E2-15%2Ax%2B7=%28x-7%29%282x-1%29\"$ .

Note: you can check the answer by expanding $\"%28x-7%29%282x-1%29\"$ to get $\"2%2Ax%5E2-15%2Ax%2B7\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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