You can
put this solution on YOUR website!#1
"Factor x^2-3x-18"
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
In order to factor  , first multiply the leading coefficient 1 and the last term -18 to get -18. Now we need to ask ourselves: What two numbers multiply to -18 and add to -3? Lets find out by listing all of the possible factors of -18
Factors:
1,2,3,6,9,18,
-1,-2,-3,-6,-9,-18, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -18.
(-1)*(18)=-18
(-2)*(9)=-18
(-3)*(6)=-18
Now which of these pairs add to -3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -3
| First Number | | | Second Number | | | Sum | | 1 | | | -18 | || | 1+(-18)=-17 | | 2 | | | -9 | || | 2+(-9)=-7 | | 3 | | | -6 | || | 3+(-6)=-3 | | -1 | | | 18 | || | (-1)+18=17 | | -2 | | | 9 | || | (-2)+9=7 | | -3 | | | 6 | || | (-3)+6=3 |
We can see from the table that 3 and -6 add to -3. So the two numbers that multiply to -18 and add to -3 are: 3 and -6
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to -18 and add to -3, which are: 3 and -6)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 1x out of the first group and factor a -6 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. |
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#2
"Factor 2y^2+8y+9y+36"
Start with the given expression
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
#3
"Factor w^2-81v^2"
Start with the given expression
Let
and
. So we get this:
Since
, A can be solved for:
Take the square root of both sides
Since
, B can be solved for:
Take the square root of both sides
Since we have a difference of squares, we can factor it like this:
Now replace A and B
Plug in
and
So the expression
factors to
Notice that if you foil the factored expression, you get the original expression. This verifies our answer.
#4
"Factor 5x^2+23xy+12y^2"
Looking at
we can see that the first term is
and the last term is
where the coefficients are 5 and -12 respectively.
Now multiply the first coefficient 5 and the last coefficient -12 to get -60. Now what two numbers multiply to -60 and add to 7? Let's list all of the factors of -60:
Factors of -60:
1,2,3,4,5,6,10,12,15,20,30,60
-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -60
(1)*(-60)
(2)*(-30)
(3)*(-20)
(4)*(-15)
(5)*(-12)
(6)*(-10)
(-1)*(60)
(-2)*(30)
(-3)*(20)
(-4)*(15)
(-5)*(12)
(-6)*(10)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7
| First Number | Second Number | Sum | | 1 | -60 | 1+(-60)=-59 |
| 2 | -30 | 2+(-30)=-28 |
| 3 | -20 | 3+(-20)=-17 |
| 4 | -15 | 4+(-15)=-11 |
| 5 | -12 | 5+(-12)=-7 |
| 6 | -10 | 6+(-10)=-4 |
| -1 | 60 | -1+60=59 |
| -2 | 30 | -2+30=28 |
| -3 | 20 | -3+20=17 |
| -4 | 15 | -4+15=11 |
| -5 | 12 | -5+12=7 |
| -6 | 10 | -6+10=4 |
From this list we can see that -5 and 12 add up to 7 and multiply to -60
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
)
(