In order to factor  , first multiply the leading coefficient 3 and the last term 5 to get 15. Now we need to ask ourselves: What two numbers multiply to 15 and add to 8? Lets find out by listing all of the possible factors of 15
Factors:
1,3,5,15,
-1,-3,-5,-15, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 15.
1*15=15
3*5=15
(-1)*(-15)=15
(-3)*(-5)=15
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 8
| First Number | | | Second Number | | | Sum | | 1 | | | 15 | || | 1+15=16 | | 3 | | | 5 | || | 3+5=8 | | -1 | | | -15 | || | -1+(-15)=-16 | | -3 | | | -5 | || | -3+(-5)=-8 |
We can see from the table that 3 and 5 add to 8. So the two numbers that multiply to 15 and add to 8 are: 3 and 5
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to 15 and add to 8, which are: 3 and 5)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 3x out of the first group and factor a 5 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. |
