SOLUTION: complete the following statements 6a^2+13a+6=(2a+3)( ) 4m^2+5mn-6n^2=(m+2n)( )

Question 117627: complete the following statements
6a^2+13a+6=(2a+3)( )
4m^2+5mn-6n^2=(m+2n)( )
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
#1

Looking at we can see that the first term is and the last term is where the coefficients are 6 and 6 respectively.

Now multiply the first coefficient 6 and the last coefficient 6 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient 13? Let's list all of the factors of 36:

Factors of 36:
1,2,3,4,6,9,12,18

-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 36
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 13

First Number	Second Number	Sum
1	36	1+36=37
2	18	2+18=20
3	12	3+12=15
4	9	4+9=13
6	6	6+6=12
-1	-36	-1+(-36)=-37
-2	-18	-2+(-18)=-20
-3	-12	-3+(-12)=-15
-4	-9	-4+(-9)=-13
-6	-6	-6+(-6)=-12

From this list we can see that 4 and 9 add up to 13 and multiply to 36

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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Answer:

So factors to which means the missing factor is

#2

Looking at we can see that the first term is and the last term is where the coefficients are 4 and -6 respectively.

Now multiply the first coefficient 4 and the last coefficient -6 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient 5? Let's list all of the factors of -24:

Factors of -24:
1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24
(1)*(-24)
(2)*(-12)
(3)*(-8)
(4)*(-6)
(-1)*(24)
(-2)*(12)
(-3)*(8)
(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to 5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 5

First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

From this list we can see that -3 and 8 add up to 5 and multiply to -24

Now looking at the expression , replace with (notice adds up to . So it is equivalent to )

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