document.write( "Question 103997This question is from textbook beginning and intermediate algebra
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Algebra.Com's Answer #75669 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 $\"2y%5E2-7y-49\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-7\"$ , and the last term is $\"-49\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"-49\"$ to get $\"%282%29%28-49%29=-98\"$ .

Now the question is: what two whole numbers multiply to $\"-98\"$ (the previous product) and add to the second coefficient $\"-7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-98\"$ (the previous product).

Factors of $\"-98\"$ :

1,2,7,14,49,98

-1,-2,-7,-14,-49,-98

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

These factors pair up and multiply to $\"-98\"$ .

1*(-98) = -98
2*(-49) = -98
7*(-14) = -98
(-1)*(98) = -98
(-2)*(49) = -98
(-7)*(14) = -98

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	-98	1+(-98)=-97
2	-49	2+(-49)=-47
7	-14	7+(-14)=-7
-1	98	-1+98=97
-2	49	-2+49=47
-7	14	-7+14=7

From the table, we can see that the two numbers $\"7\"$ and $\"-14\"$ add to $\"-7\"$ (the middle coefficient).

So the two numbers $\"7\"$ and $\"-14\"$ both multiply to $\"-98\"$ and add to $\"-7\"$

Now replace the middle term $\"-7y\"$ with $\"7y-14y\"$ . Remember, $\"7\"$ and $\"-14\"$ add to $\"-7\"$ . So this shows us that $\"7y-14y=-7y\"$ .

$\"2y%5E2%2Bhighlight%287y-14y%29-49\"$ Replace the second term $\"-7y\"$ with $\"7y-14y\"$ .

$\"%282y%5E2%2B7y%29%2B%28-14y-49%29\"$ Group the terms into two pairs.

$\"y%282y%2B7%29%2B%28-14y-49%29\"$ Factor out the GCF $\"y\"$ from the first group.

$\"y%282y%2B7%29-7%282y%2B7%29\"$ 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.

$\"%28y-7%29%282y%2B7%29\"$ Combine like terms. Or factor out the common term $\"2y%2B7\"$

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Answer:

So $\"2%2Ay%5E2-7%2Ay-49\"$ factors to $\"%28y-7%29%282y%2B7%29\"$ .

In other words, $\"2%2Ay%5E2-7%2Ay-49=%28y-7%29%282y%2B7%29\"$ .

Note: you can check the answer by expanding $\"%28y-7%29%282y%2B7%29\"$ to get $\"2%2Ay%5E2-7%2Ay-49\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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