Question 117829: Rewrite the middle term as the sum of two terms and then factor completely.
10x^2+19x+6
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 10 and 6 respectively.
Now multiply the first coefficient 10 and the last coefficient 6 to get 60. Now what two numbers multiply to 60 and add to the middle coefficient 19? Let's list all of the factors of 60:
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 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 60
1*60
2*30
3*20
4*15
5*12
6*10
(-1)*(-60)
(-2)*(-30)
(-3)*(-20)
(-4)*(-15)
(-5)*(-12)
(-6)*(-10)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19
| First Number | Second Number | Sum | | 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 | | -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 this list we can see that 4 and 15 add up to 19 and multiply to 60
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of  out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
So this also means that factors to (since is equivalent to )
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Answer:
So factors to
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