SOLUTION: Factor 9x² -6x -8

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Question 244621: Factor 9x² -6x -8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 9x%5E2-6x-8, we can see that the first coefficient is 9, the second coefficient is -6, and the last term is -8.


Now multiply the first coefficient 9 by the last term -8 to get %289%29%28-8%29=-72.


Now the question is: what two whole numbers multiply to -72 (the previous product) and add to the second coefficient -6?


To find these two numbers, we need to list all of the factors of -72 (the previous product).


Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72


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


These factors pair up and multiply to -72.
1*(-72) = -72
2*(-36) = -72
3*(-24) = -72
4*(-18) = -72
6*(-12) = -72
8*(-9) = -72
(-1)*(72) = -72
(-2)*(36) = -72
(-3)*(24) = -72
(-4)*(18) = -72
(-6)*(12) = -72
(-8)*(9) = -72

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -6:


First NumberSecond NumberSum
1-721+(-72)=-71
2-362+(-36)=-34
3-243+(-24)=-21
4-184+(-18)=-14
6-126+(-12)=-6
8-98+(-9)=-1
-172-1+72=71
-236-2+36=34
-324-3+24=21
-418-4+18=14
-612-6+12=6
-89-8+9=1



From the table, we can see that the two numbers 6 and -12 add to -6 (the middle coefficient).


So the two numbers 6 and -12 both multiply to -72 and add to -6


Now replace the middle term -6x with 6x-12x. Remember, 6 and -12 add to -6. So this shows us that 6x-12x=-6x.


9x%5E2%2Bhighlight%286x-12x%29-8 Replace the second term -6x with 6x-12x.


%289x%5E2%2B6x%29%2B%28-12x-8%29 Group the terms into two pairs.


3x%283x%2B2%29%2B%28-12x-8%29 Factor out the GCF 3x from the first group.


3x%283x%2B2%29-4%283x%2B2%29 Factor out 4 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.


%283x-4%29%283x%2B2%29 Combine like terms. Or factor out the common term 3x%2B2


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Answer:


So 9x%5E2-6x-8 factors to %283x-4%29%283x%2B2%29.


In other words, 9x%5E2-6x-8=%283x-4%29%283x%2B2%29.


Note: you can check the answer by expanding %283x-4%29%283x%2B2%29 to get 9x%5E2-6x-8 or by graphing the original expression and the answer (the two graphs should be identical).