You can put this solution on YOUR website! In order to factor , first multiply 12 and 4 to get 48 and we need to ask ourselves: What two numbers multiply to 48 and add to 19? Lets find out by listing all of the possible factors of 48
Factors:
1,2,3,4,6,8,12,16,24,48,
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to 48.
1*48=48
2*24=48
3*16=48
4*12=48
6*8=48
(-1)*(-48)=48
(-2)*(-24)=48
(-3)*(-16)=48
(-4)*(-12)=48
(-6)*(-8)=48
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to 19? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 19
First Number
|
Second Number
|
Sum
1
|
48
|
|
1+48=49
2
|
24
|
|
2+24=26
3
|
16
|
|
3+16=19
4
|
12
|
|
4+12=16
6
|
8
|
|
6+8=14
-1
|
-48
|
|
-1+(-48)=-49
-2
|
-24
|
|
-2+(-24)=-26
-3
|
-16
|
|
-3+(-16)=-19
-4
|
-12
|
|
-4+(-12)=-16
-6
|
-8
|
|
-6+(-8)=-14
We can see from the table that 3 and 16 add to 19. So the two numbers that multiply to 48 and add to 19 are: 3 and 16
So the original quadratic
breaks down to this (just replace with the two numbers that multiply to 48 and add to 19, which are: 3 and 16)
Group the first two terms together and the last two terms together like this:
Factor a 3 out of the first group and factor a 4 out of the second group.
Now since we have a common term we can combine the two terms. Notice if we let we would get . Since we have that common term , we are able to combine and
Combine like terms.
Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer.
Now lets factor the quadratic inside the parenthesis
In order to factor , first multiply 4 and -10 to get -40 and we need to ask ourselves: What two numbers multiply to -40 and add to -18? Lets find out by listing all of the possible factors of -40
Factors:
1,2,4,5,8,10,20,40,
-1,-2,-4,-5,-8,-10,-20,-40, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -40.
(-1)*(40)=-40
(-2)*(20)=-40
(-4)*(10)=-40
(-5)*(8)=-40
Now which of these pairs add to -18? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -18
First Number
|
Second Number
|
Sum
1
|
-40
|
|
1+(-40)=-39
2
|
-20
|
|
2+(-20)=-18
4
|
-10
|
|
4+(-10)=-6
5
|
-8
|
|
5+(-8)=-3
-1
|
40
|
|
(-1)+40=39
-2
|
20
|
|
(-2)+20=18
-4
|
10
|
|
(-4)+10=6
-5
|
8
|
|
(-5)+8=3
We can see from the table that 2 and -20 add to -18. So the two numbers that multiply to -40 and add to -18 are: 2 and -20
So the original quadratic
breaks down to this (just replace with the two numbers that multiply to -40 and add to -18, which are: 2 and -20)
Group the first two terms together and the last two terms together like this:
Factor a 2 out of the first group and factor a -10 out of the second group.
Now since we have a common term we can combine the two terms. Notice if we let we would get . Since we have that common term , we are able to combine and
Combine like terms.
Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer.
So now reintroduce the z term we factored out back in:
(