document.write( "Question 264019: can you help me with this question:
\n" ); document.write( "factor the trimonial a to the second+2a-63 thanks \n" ); document.write( "

Algebra.Com's Answer #194458 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 $\"a%5E2%2B2a-63\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-63\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-63\"$ to get $\"%281%29%28-63%29=-63\"$ .

Now the question is: what two whole numbers multiply to $\"-63\"$ (the previous product) and add to the second coefficient $\"2\"$ ?

To find these two numbers, we need to list all of the factors of $\"-63\"$ (the previous product).

Factors of $\"-63\"$ :

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63

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

These factors pair up and multiply to $\"-63\"$ .

1*(-63) = -63
3*(-21) = -63
7*(-9) = -63
(-1)*(63) = -63
(-3)*(21) = -63
(-7)*(9) = -63

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"2\"$ :

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First Number	Second Number	Sum
1	-63	1+(-63)=-62
3	-21	3+(-21)=-18
7	-9	7+(-9)=-2
-1	63	-1+63=62
-3	21	-3+21=18
-7	9	-7+9=2

From the table, we can see that the two numbers $\"-7\"$ and $\"9\"$ add to $\"2\"$ (the middle coefficient).

So the two numbers $\"-7\"$ and $\"9\"$ both multiply to $\"-63\"$ and add to $\"2\"$

Now replace the middle term $\"2a\"$ with $\"-7a%2B9a\"$ . Remember, $\"-7\"$ and $\"9\"$ add to $\"2\"$ . So this shows us that $\"-7a%2B9a=2a\"$ .

$\"a%5E2%2Bhighlight%28-7a%2B9a%29-63\"$ Replace the second term $\"2a\"$ with $\"-7a%2B9a\"$ .

$\"%28a%5E2-7a%29%2B%289a-63%29\"$ Group the terms into two pairs.

$\"a%28a-7%29%2B%289a-63%29\"$ Factor out the GCF $\"a\"$ from the first group.

$\"a%28a-7%29%2B9%28a-7%29\"$ Factor out $\"9\"$ 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.

$\"%28a%2B9%29%28a-7%29\"$ Combine like terms. Or factor out the common term $\"a-7\"$

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Answer:

So $\"a%5E2%2B2%2Aa-63\"$ factors to $\"%28a%2B9%29%28a-7%29\"$ .

In other words, $\"a%5E2%2B2%2Aa-63=%28a%2B9%29%28a-7%29\"$ .

Note: you can check the answer by expanding $\"%28a%2B9%29%28a-7%29\"$ to get $\"a%5E2%2B2%2Aa-63\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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