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Algebra.Com's Answer #191665 by richwmiller(17219) $\"\"$ $\"About$

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

Looking at the expression $\"4x%5E2%2B0x-49\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"0\"$ , and the last term is $\"-49\"$ .

Now multiply the first coefficient $\"4\"$ by the last term $\"-49\"$ to get $\"%284%29%28-49%29=-196\"$ .

Now the question is: what two whole numbers multiply to $\"-196\"$ (the previous product) and add to the second coefficient $\"0\"$ ?

To find these two numbers, we need to list all of the factors of $\"-196\"$ (the previous product).

Factors of $\"-196\"$ :

1,2,4,7,14,28,49,98,196

-1,-2,-4,-7,-14,-28,-49,-98,-196

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

These factors pair up and multiply to $\"-196\"$ .

1*(-196) = -196
2*(-98) = -196
4*(-49) = -196
7*(-28) = -196
14*(-14) = -196
(-1)*(196) = -196
(-2)*(98) = -196
(-4)*(49) = -196
(-7)*(28) = -196
(-14)*(14) = -196

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"0\"$ :

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First Number	Second Number	Sum
1	-196	1+(-196)=-195
2	-98	2+(-98)=-96
4	-49	4+(-49)=-45
7	-28	7+(-28)=-21
14	-14	14+(-14)=0
-1	196	-1+196=195
-2	98	-2+98=96
-4	49	-4+49=45
-7	28	-7+28=21
-14	14	-14+14=0

From the table, we can see that the two numbers $\"-14\"$ and $\"14\"$ add to $\"0\"$ (the middle coefficient).

So the two numbers $\"-14\"$ and $\"14\"$ both multiply to $\"-196\"$ and add to $\"0\"$

Now replace the middle term $\"0x\"$ with $\"-14x%2B14x\"$ . Remember, $\"-14\"$ and $\"14\"$ add to $\"0\"$ . So this shows us that $\"-14x%2B14x=0x\"$ .

$\"4x%5E2%2Bhighlight%28-14x%2B14x%29-49\"$ Replace the second term $\"0x\"$ with $\"-14x%2B14x\"$ .

$\"%284x%5E2-14x%29%2B%2814x-49%29\"$ Group the terms into two pairs.

$\"2x%282x-7%29%2B%2814x-49%29\"$ Factor out the GCF $\"2x\"$ from the first group.

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

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

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Answer:

So $\"4%2Ax%5E2%2B0%2Ax-49\"$ factors to $\"%282x%2B7%29%282x-7%29\"$ .

In other words, $\"4%2Ax%5E2%2B0%2Ax-49=%282x%2B7%29%282x-7%29\"$ .

Note: you can check the answer by expanding $\"%282x%2B7%29%282x-7%29\"$ to get $\"4%2Ax%5E2%2B0%2Ax-49\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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