SOLUTION: How do I factor 6n^2-78n+216

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Question 368783: How do I factor 6n^2-78n+216
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
6n%5E2-78n%2B216 Start with the given expression.


6%28n%5E2-13n%2B36%29 Factor out the GCF 6.


Now let's try to factor the inner expression n%5E2-13n%2B36


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Looking at the expression n%5E2-13n%2B36, we can see that the first coefficient is 1, the second coefficient is -13, and the last term is 36.


Now multiply the first coefficient 1 by the last term 36 to get %281%29%2836%29=36.


Now the question is: what two whole numbers multiply to 36 (the previous product) and add to the second coefficient -13?


To find these two numbers, we need to list all of the factors of 36 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -13:


First NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+6=12
-1-36-1+(-36)=-37
-2-18-2+(-18)=-20
-3-12-3+(-12)=-15
-4-9-4+(-9)=-13
-6-6-6+(-6)=-12



From the table, we can see that the two numbers -4 and -9 add to -13 (the middle coefficient).


So the two numbers -4 and -9 both multiply to 36 and add to -13


Now replace the middle term -13n with -4n-9n. Remember, -4 and -9 add to -13. So this shows us that -4n-9n=-13n.


n%5E2%2Bhighlight%28-4n-9n%29%2B36 Replace the second term -13n with -4n-9n.


%28n%5E2-4n%29%2B%28-9n%2B36%29 Group the terms into two pairs.


n%28n-4%29%2B%28-9n%2B36%29 Factor out the GCF n from the first group.


n%28n-4%29-9%28n-4%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.


%28n-9%29%28n-4%29 Combine like terms. Or factor out the common term n-4


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So 6%28n%5E2-13n%2B36%29 then factors further to 6%28n-9%29%28n-4%29


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Answer:


So 6n%5E2-78n%2B216 completely factors to 6%28n-9%29%28n-4%29.


In other words, 6n%5E2-78n%2B216=6%28n-9%29%28n-4%29.


Note: you can check the answer by expanding 6%28n-9%29%28n-4%29 to get 6n%5E2-78n%2B216 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim