SOLUTION: Factor 6x^2-13x+6

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Question 370520: Factor 6x^2-13x+6
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression 6x%5E2-13x%2B6, we can see that the first coefficient is 6, the second coefficient is -13, and the last term is 6.


Now multiply the first coefficient 6 by the last term 6 to get %286%29%286%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 -13x with -4x-9x. Remember, -4 and -9 add to -13. So this shows us that -4x-9x=-13x.


6x%5E2%2Bhighlight%28-4x-9x%29%2B6 Replace the second term -13x with -4x-9x.


%286x%5E2-4x%29%2B%28-9x%2B6%29 Group the terms into two pairs.


2x%283x-2%29%2B%28-9x%2B6%29 Factor out the GCF 2x from the first group.


2x%283x-2%29-3%283x-2%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.


%282x-3%29%283x-2%29 Combine like terms. Or factor out the common term 3x-2


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Answer:


So 6x%5E2-13x%2B6 factors to %282x-3%29%283x-2%29.


In other words, 6x%5E2-13x%2B6=%282x-3%29%283x-2%29.


Note: you can check the answer by expanding %282x-3%29%283x-2%29 to get 6x%5E2-13x%2B6 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