SOLUTION: 6x2 - 28x + 16 (Factor Competely)

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Question 148772: 6x2 - 28x + 16 (Factor Competely)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6x%5E2-28x%2B16 Start with the given expression


2%283x%5E2-14x%2B8%29 Factor out the GCF 2


Now let's focus on the inner expression 3x%5E2-14x%2B8



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Looking at the expression 3x%5E2-14x%2B8, we can see that the first coefficient is 3, the second coefficient is -14, and the last term is 8.


Now multiply the first coefficient 3 by the last term 8 to get %283%29%288%29=24.


Now the question is: what two whole numbers multiply to 24 (the previous product) and add to the second coefficient -14?


To find these two numbers, we need to list all of the factors of 24 (the previous product).


Factors of 24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


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


These factors pair up and multiply to 24. For instance, 1%2A24=24, 2%2A12=24, etc.


Since 24 is positive, this means that either
a) both factors are positive, or...
b) both factors are negative.


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -14:


First NumberSecond NumberSum
1241+24=25
2122+12=14
383+8=11
464+6=10
-1-24-1+(-24)=-25
-2-12-2+(-12)=-14
-3-8-3+(-8)=-11
-4-6-4+(-6)=-10



From the table, we can see that the two numbers -2 and -12 add to -14 (the middle coefficient).


So the two numbers -2 and -12 both multiply to 24 and add to -14


Now replace the middle term -14x with -2x-12x. Remember, -2 and -12 add to -14. So this shows us that -2x-12x=-14x.


3x%5E2%2Bhighlight%28-2x-12x%29%2B8 Replace the second term -14x with -2x-12x.


%283x%5E2-2x%29%2B%28-12x%2B8%29 Group the terms into two pairs.


x%283x-2%29%2B%28-12x%2B8%29 Factor out the GCF x from the first group.


x%283x-2%29-4%283x-2%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.


%28x-4%29%283x-2%29 Combine like terms. Or factor out the common term 3x-2





So 3x%5E2-14x%2B8 factors to %28x-4%29%283x-2%29.


So this means that 2%283x%5E2-14x%2B8%29 can be rewritten as 2%28x-4%29%283x-2%29



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Answer:



So 6x%5E2-28x%2B16 factors to 2%28x-4%29%283x-2%29.



Note: you can check the answer by expanding 2%28x-4%29%283x-2%29 to get 6x%5E2-28x%2B16 or by graphing the original expression and the answer (the two graphs should be identical).