|
Question 176941: Hello wonderful ones. I have yet another problem.
How would you factor
3x^2+22xy+24y^2
Your help is so appreciated.
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 3 and 24 respectively.
Now multiply the first coefficient 3 and the last coefficient 24 to get 72. Now what two numbers multiply to 72 and add to the middle coefficient 22? 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 two negative numbers multiplied together make a positive number
Now which of these pairs add to 22? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 22
| First Number | Second Number | Sum | | 1 | 72 | 1+72=73 | | 2 | 36 | 2+36=38 | | 3 | 24 | 3+24=27 | | 4 | 18 | 4+18=22 | | 6 | 12 | 6+12=18 | | 8 | 9 | 8+9=17 | | -1 | -72 | -1+(-72)=-73 | | -2 | -36 | -2+(-36)=-38 | | -3 | -24 | -3+(-24)=-27 | | -4 | -18 | -4+(-18)=-22 | | -6 | -12 | -6+(-12)=-18 | | -8 | -9 | -8+(-9)=-17 |
From this list we can see that 4 and 18 add up to 22 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 )
------------------------------------------------------------
Answer:
So factors to
|
|
|
| |
