In order to factor  , first multiply the leading coefficient 3 and the last term -36 to get -108. Now we need to ask ourselves: What two numbers multiply to -108 and add to 23? Lets find out by listing all of the possible factors of -108
Factors:
1,2,3,4,6,9,12,18,27,36,54,108,
-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-108, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -108.
(-1)*(108)=-108
(-2)*(54)=-108
(-3)*(36)=-108
(-4)*(27)=-108
(-6)*(18)=-108
(-9)*(12)=-108
Now which of these pairs add to 23? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 23
| First Number | | | Second Number | | | Sum | | 1 | | | -108 | || | 1+(-108)=-107 | | 2 | | | -54 | || | 2+(-54)=-52 | | 3 | | | -36 | || | 3+(-36)=-33 | | 4 | | | -27 | || | 4+(-27)=-23 | | 6 | | | -18 | || | 6+(-18)=-12 | | 9 | | | -12 | || | 9+(-12)=-3 | | -1 | | | 108 | || | (-1)+108=107 | | -2 | | | 54 | || | (-2)+54=52 | | -3 | | | 36 | || | (-3)+36=33 | | -4 | | | 27 | || | (-4)+27=23 | | -6 | | | 18 | || | (-6)+18=12 | | -9 | | | 12 | || | (-9)+12=3 |
We can see from the table that -4 and 27 add to 23. So the two numbers that multiply to -108 and add to 23 are: -4 and 27
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to -108 and add to 23, which are: -4 and 27)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 1 out of the first group and factor a 9 out of the second group.
Now since we have a common term  we can combine the two terms.
 Combine like terms.
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Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer. |
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