document.write( "Question 254996: How do I factor x^2+3x-54? Plz help!!! \n" ); document.write( "

Algebra.Com's Answer #187296 by richwmiller(17219) $\"\"$ $\"About$

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

Looking at the expression $\"x%5E2%2B3x-54\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-54\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-54\"$ to get $\"%281%29%28-54%29=-54\"$ .

Now the question is: what two whole numbers multiply to $\"-54\"$ (the previous product) and add to the second coefficient $\"3\"$ ?

To find these two numbers, we need to list all of the factors of $\"-54\"$ (the previous product).

Factors of $\"-54\"$ :

1,2,3,6,9,18,27,54

-1,-2,-3,-6,-9,-18,-27,-54

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

These factors pair up and multiply to $\"-54\"$ .

1*(-54) = -54
2*(-27) = -54
3*(-18) = -54
6*(-9) = -54
(-1)*(54) = -54
(-2)*(27) = -54
(-3)*(18) = -54
(-6)*(9) = -54

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"3\"$ :

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First Number	Second Number	Sum
1	-54	1+(-54)=-53
2	-27	2+(-27)=-25
3	-18	3+(-18)=-15
6	-9	6+(-9)=-3
-1	54	-1+54=53
-2	27	-2+27=25
-3	18	-3+18=15
-6	9	-6+9=3

From the table, we can see that the two numbers $\"-6\"$ and $\"9\"$ add to $\"3\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"9\"$ both multiply to $\"-54\"$ and add to $\"3\"$

Now replace the middle term $\"3x\"$ with $\"-6x%2B9x\"$ . Remember, $\"-6\"$ and $\"9\"$ add to $\"3\"$ . So this shows us that $\"-6x%2B9x=3x\"$ .

$\"x%5E2%2Bhighlight%28-6x%2B9x%29-54\"$ Replace the second term $\"3x\"$ with $\"-6x%2B9x\"$ .

$\"%28x%5E2-6x%29%2B%289x-54%29\"$ Group the terms into two pairs.

$\"x%28x-6%29%2B%289x-54%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-6%29%2B9%28x-6%29\"$ 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.

$\"%28x%2B9%29%28x-6%29\"$ Combine like terms. Or factor out the common term $\"x-6\"$

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Answer:

So $\"x%5E2%2B3%2Ax-54\"$ factors to $\"%28x%2B9%29%28x-6%29\"$ .

In other words, $\"x%5E2%2B3%2Ax-54=%28x%2B9%29%28x-6%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B9%29%28x-6%29\"$ to get $\"x%5E2%2B3%2Ax-54\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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