SOLUTION: Factor completely. If it is prime, say so. 9x^2-9x-54

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Question 109539: Factor completely. If it is prime, say so.
9x^2-9x-54
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


9%2Ax%5E2-9%2Ax-54 Start with the given expression.



9%28x%5E2-x-6%29 Factor out the GCF 9.



Now let's try to factor the inner expression x%5E2-x-6



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Looking at the expression x%5E2-x-6, we can see that the first coefficient is 1, the second coefficient is -1, and the last term is -6.



Now multiply the first coefficient 1 by the last term -6 to get %281%29%28-6%29=-6.



Now the question is: what two whole numbers multiply to -6 (the previous product) and add to the second coefficient -1?



To find these two numbers, we need to list all of the factors of -6 (the previous product).



Factors of -6:

1,2,3,6

-1,-2,-3,-6



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



These factors pair up and multiply to -6.

1*(-6) = -6
2*(-3) = -6
(-1)*(6) = -6
(-2)*(3) = -6


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:



First NumberSecond NumberSum
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1




From the table, we can see that the two numbers 2 and -3 add to -1 (the middle coefficient).



So the two numbers 2 and -3 both multiply to -6 and add to -1



Now replace the middle term -1x with 2x-3x. Remember, 2 and -3 add to -1. So this shows us that 2x-3x=-1x.



x%5E2%2Bhighlight%282x-3x%29-6 Replace the second term -1x with 2x-3x.



%28x%5E2%2B2x%29%2B%28-3x-6%29 Group the terms into two pairs.



x%28x%2B2%29%2B%28-3x-6%29 Factor out the GCF x from the first group.



x%28x%2B2%29-3%28x%2B2%29 Factor out 3 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-3%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2



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So 9%28x%5E2-x-6%29 then factors further to 9%28x-3%29%28x%2B2%29



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Answer:



So 9%2Ax%5E2-9%2Ax-54 completely factors to 9%28x-3%29%28x%2B2%29.



In other words, 9%2Ax%5E2-9%2Ax-54=9%28x-3%29%28x%2B2%29.



Note: you can check the answer by expanding 9%28x-3%29%28x%2B2%29 to get 9%2Ax%5E2-9%2Ax-54 or by graphing the original expression and the answer (the two graphs should be identical).