SOLUTION: 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

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

===============================================================

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).

#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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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