SOLUTION: HOW DO I FACTOR PROBLEMS LIKE X^2-9XY+14Y^2 OR 6a^2-5ab-4b^2 please show the work for it thanks

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Question 195512: HOW DO I FACTOR PROBLEMS LIKE
X^2-9XY+14Y^2
OR
6a^2-5ab-4b^2
please show the work for it
thanks

Answer by jim_thompson5910(28595)   (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 1 and 14 respectively.

Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the middle coefficient -9? Let's list all of the factors of 14:

Factors of 14:
1,2,7,14

-1,-2,-7,-14 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 14
1*14
2*7
(-1)*(-14)
(-2)*(-7)

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

Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9

First NumberSecond NumberSum
1141+14=15
272+7=9
-1-14-1+(-14)=-15
-2-7-2+(-7)=-9

From this list we can see that -2 and -7 add up to -9 and multiply to 14

Now looking at the expression , replace with (notice combines 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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So factors to

# 2

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

Now multiply the first coefficient 6 and the last coefficient -4 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 , 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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