Question 110760: Hello can you answer these questions for me1
1. Factor
x^2-3x-18
2. Factor
2y^2+8y+9y+36
3. Factor
w^2-81v^2
4. Factor
5x^2+23xy+12y^2
Found 2 solutions by scott8148, jim_thompson5910:
Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website! 1. (x-6)(x+3)
2. (2y+9)(y+4)
3. difference of two perfect squares factors into sum and difference of square roots ... (w+9v)(w-9v)
4. (5x+3y)(x+4y)
Answer by jim_thompson5910(35256)  (Show Source):
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) |
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| 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 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.
 Combine like terms. Or factor out the common term 
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Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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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 )
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