Question 112001: Here's another I can't factor:
(-8k^2 + 6K + 9)
thanks very much,
CJ
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 -8 and 9 respectively.
Now multiply the first coefficient -8 and the last coefficient 9 to get -72. Now what two numbers multiply to -72 and add to 6? Let's list all of the factors of -72:
Factors of -72:
1,2,3,4,6,8,9,12,18,24,36,72
-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -72
(1)*(-72)
(2)*(-36)
(3)*(-24)
(4)*(-18)
(6)*(-12)
(8)*(-9)
(-1)*(72)
(-2)*(36)
(-3)*(24)
(-4)*(18)
(-6)*(12)
(-8)*(9)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 6
| First Number | Second Number | Sum | | 1 | -72 | 1+(-72)=-71 | | 2 | -36 | 2+(-36)=-34 | | 3 | -24 | 3+(-24)=-21 | | 4 | -18 | 4+(-18)=-14 | | 6 | -12 | 6+(-12)=-6 | | 8 | -9 | 8+(-9)=-1 | | -1 | 72 | -1+72=71 | | -2 | 36 | -2+36=34 | | -3 | 24 | -3+24=21 | | -4 | 18 | -4+18=14 | | -6 | 12 | -6+12=6 | | -8 | 9 | -8+9=1 |
From this list we can see that -6 and 12 add up to 6 and multiply to -72
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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