In order to factor  , first multiply the leading coefficient 8 and the last term 24 to get 192. Now we need to ask ourselves: What two numbers multiply to 192 and add to -32? Lets find out by listing all of the possible factors of 192
Factors:
1,2,3,4,6,8,12,16,24,32,48,64,96,192,
-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-64,-96,-192, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 192.
1*192=192
2*96=192
3*64=192
4*48=192
6*32=192
8*24=192
12*16=192
(-1)*(-192)=192
(-2)*(-96)=192
(-3)*(-64)=192
(-4)*(-48)=192
(-6)*(-32)=192
(-8)*(-24)=192
(-12)*(-16)=192
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -32? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -32
| First Number | | | Second Number | | | Sum | | 1 | | | 192 | || | 1+192=193 | | 2 | | | 96 | || | 2+96=98 | | 3 | | | 64 | || | 3+64=67 | | 4 | | | 48 | || | 4+48=52 | | 6 | | | 32 | || | 6+32=38 | | 8 | | | 24 | || | 8+24=32 | | 12 | | | 16 | || | 12+16=28 | | -1 | | | -192 | || | -1+(-192)=-193 | | -2 | | | -96 | || | -2+(-96)=-98 | | -3 | | | -64 | || | -3+(-64)=-67 | | -4 | | | -48 | || | -4+(-48)=-52 | | -6 | | | -32 | || | -6+(-32)=-38 | | -8 | | | -24 | || | -8+(-24)=-32 | | -12 | | | -16 | || | -12+(-16)=-28 |
We can see from the table that -8 and -24 add to -32. So the two numbers that multiply to 192 and add to -32 are: -8 and -24
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to 192 and add to -32, which are: -8 and -24)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 8 out of the first group and factor a -24 out of the second group.
Now since we have a common term  we can combine the two terms.
 Combine like terms.
==============================================================================
Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer. |
(