SOLUTION: Hello wonderful ones. I have yet another problem. How would you factor 3x^2+22xy+24y^2 Your help is so appreciated.

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

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