SOLUTION: I have to factor completely 10-19c+6c^2 I am having issues trying to find a common factor...

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Question 741856: I have to factor completely
10-19c+6c^2
I am having issues trying to find a common factor...

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
There is no common factor (other than 1), so you have do to this

10-19c%2B6c%5E2

is the same as

6c%5E2-19c%2B10

Now let's factor



Looking at the expression 6c%5E2-19c%2B10, we can see that the first coefficient is 6, the second coefficient is -19, and the last term is 10.


Now multiply the first coefficient 6 by the last term 10 to get %286%29%2810%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient -19?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


Factors of 60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60


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


These factors pair up and multiply to 60.
1*60 = 60
2*30 = 60
3*20 = 60
4*15 = 60
5*12 = 60
6*10 = 60
(-1)*(-60) = 60
(-2)*(-30) = 60
(-3)*(-20) = 60
(-4)*(-15) = 60
(-5)*(-12) = 60
(-6)*(-10) = 60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -19:


First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From the table, we can see that the two numbers -4 and -15 add to -19 (the middle coefficient).


So the two numbers -4 and -15 both multiply to 60 and add to -19


Now replace the middle term -19c with -4c-15c. Remember, -4 and -15 add to -19. So this shows us that -4c-15c=-19c.


6c%5E2%2Bhighlight%28-4c-15c%29%2B10 Replace the second term -19c with -4c-15c.


%286c%5E2-4c%29%2B%28-15c%2B10%29 Group the terms into two pairs.


2c%283c-2%29%2B%28-15c%2B10%29 Factor out the GCF 2c from the first group.


2c%283c-2%29-5%283c-2%29 Factor out 5 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.


%282c-5%29%283c-2%29 Combine like terms. Or factor out the common term 3c-2


===============================================================


Answer:


So 6c%5E2-19c%2B10 factors to %282c-5%29%283c-2%29.


In other words, 6c%5E2-19c%2B10=%282c-5%29%283c-2%29.


Note: you can check the answer by expanding %282c-5%29%283c-2%29 to get 6c%5E2-19c%2B10 or by graphing the original expression and the answer (the two graphs should be identical).