Question 114220: Factor completely.
14x squared - 20x + 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 14 and 6 respectively.
Now multiply the first coefficient 14 and the last coefficient 6 to get 84. Now what two numbers multiply to 84 and add to the middle coefficient -20? Let's list all of the factors of 84:
Factors of 84:
1,2,3,4,6,7,12,14,21,28,42,84
-1,-2,-3,-4,-6,-7,-12,-14,-21,-28,-42,-84 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 84
1*84
2*42
3*28
4*21
6*14
7*12
(-1)*(-84)
(-2)*(-42)
(-3)*(-28)
(-4)*(-21)
(-6)*(-14)
(-7)*(-12)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -20
| First Number | Second Number | Sum | | 1 | 84 | 1+84=85 | | 2 | 42 | 2+42=44 | | 3 | 28 | 3+28=31 | | 4 | 21 | 4+21=25 | | 6 | 14 | 6+14=20 | | 7 | 12 | 7+12=19 | | -1 | -84 | -1+(-84)=-85 | | -2 | -42 | -2+(-42)=-44 | | -3 | -28 | -3+(-28)=-31 | | -4 | -21 | -4+(-21)=-25 | | -6 | -14 | -6+(-14)=-20 | | -7 | -12 | -7+(-12)=-19 |
From this list we can see that -6 and -14 add up to -20 and multiply to 84
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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