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

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


   

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

Looking at 6a%5E2%2B13a%2B6 we can see that the first term is 6a%5E2 and the last term is 6 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 NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+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 6a%5E2%2B13a%2B6, replace 13a with 4a%2B9a (notice 4a%2B9a adds up to 13a. So it is equivalent to 13a)

6a%5E2%2Bhighlight%284a%2B9a%29%2B6


Now let's factor 6a%5E2%2B4a%2B9a%2B6 by grouping:


%286a%5E2%2B4a%29%2B%289a%2B6%29 Group like terms


2a%283a%2B2%29%2B3%283a%2B2%29 Factor out the GCF of 2a out of the first group. Factor out the GCF of 3 out of the second group


%282a%2B3%29%283a%2B2%29 Since we have a common term of 3a%2B2, we can combine like terms

So 6a%5E2%2B4a%2B9a%2B6 factors to %282a%2B3%29%283a%2B2%29


So this also means that 6a%5E2%2B13a%2B6 factors to %282a%2B3%29%283a%2B2%29 (since 6a%5E2%2B13a%2B6 is equivalent to 6a%5E2%2B4a%2B9a%2B6)

-------------------------------
Answer:

So 6a%5E2%2B13a%2B6 factors to %282a%2B3%29%283a%2B2%29 which means the missing factor is 3a%2B2





#2



Looking at 4m%5E2%2B5mn-6n%5E2 we can see that the first term is 4m%5E2 and the last term is -6n%5E2 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 NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+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 4m%5E2%2B5mn-6n%5E2, replace 5mn with -3mn%2B8mn (notice -3mn%2B8mn adds up to 5mn. So it is equivalent to 5mn)

4m%5E2%2Bhighlight%28-3mn%2B8mn%29%2B-6n%5E2


Now let's factor 4m%5E2-3mn%2B8mn-6n%5E2 by grouping:


%284m%5E2-3mn%29%2B%288mn-6n%5E2%29 Group like terms


m%284m-3n%29%2B2n%284m-3n%29 Factor out the GCF of m out of the first group. Factor out the GCF of 2n out of the second group


%28m%2B2n%29%284m-3n%29 Since we have a common term of 4m-3n, we can combine like terms

So 4m%5E2-3mn%2B8mn-6n%5E2 factors to %28m%2B2n%29%284m-3n%29


So this also means that 4m%5E2%2B5mn-6n%5E2 factors to %28m%2B2n%29%284m-3n%29 (since 4m%5E2%2B5mn-6n%5E2 is equivalent to 4m%5E2-3mn%2B8mn-6n%5E2)

-------------------------------
Answer:

So 4m%5E2%2B5mn-6n%5E2 factors to %28m%2B2n%29%284m-3n%29 which means the missing factor is 4m-3n