In order to factor  , first multiply the leading coefficient 4 and the last term 9 to get 36. Now we need to ask ourselves: What two numbers multiply to 36 and add to -12? Lets find out by listing all of the possible factors of 36
Factors:
1,2,3,4,6,9,12,18,36,
-1,-2,-3,-4,-6,-9,-12,-18,-36, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 36.
1*36=36
2*18=36
3*12=36
4*9=36
6*6=36
(-1)*(-36)=36
(-2)*(-18)=36
(-3)*(-12)=36
(-4)*(-9)=36
(-6)*(-6)=36
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12
| First Number | | | Second Number | | | Sum | | 1 | | | 36 | || | 1+36=37 | | 2 | | | 18 | || | 2+18=20 | | 3 | | | 12 | || | 3+12=15 | | 4 | | | 9 | || | 4+9=13 | | 6 | | | 6 | || | 6+6=12 | | -1 | | | -36 | || | -1+(-36)=-37 | | -2 | | | -18 | || | -2+(-18)=-20 | | -3 | | | -12 | || | -3+(-12)=-15 | | -4 | | | -9 | || | -4+(-9)=-13 | | -6 | | | -6 | || | -6+(-6)=-12 |
We can see from the table that -6 and -6 add to -12. So the two numbers that multiply to 36 and add to -12 are: -6 and -6
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to 36 and add to -12, which are: -6 and -6)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 2x out of the first group and factor a -3 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
which can also be written as  since the factors repeat themselves
Notice how foils back to our original problem . This verifies our answer. |
