SOLUTION: Factor 5a^2-2ab-7b^2

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Question 114549: Factor 5a^2-2ab-7b^2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 5a%5E2-2ab-7b%5E2 we can see that the first term is 5a%5E2 and the last term is -7b%5E2 where the coefficients are 5 and -7 respectively.

Now multiply the first coefficient 5 and the last coefficient -7 to get -35. Now what two numbers multiply to -35 and add to the middle coefficient -2? Let's list all of the factors of -35:



Factors of -35:
1,5,7,35

-1,-5,-7,-35 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -35
(1)*(-35)
(5)*(-7)
(-1)*(35)
(-5)*(7)

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


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
1-351+(-35)=-34
5-75+(-7)=-2
-135-1+35=34
-57-5+7=2



From this list we can see that 5 and -7 add up to -2 and multiply to -35


Now looking at the expression 5a%5E2-2ab-7b%5E2, replace -2ab with 5ab%2B-7ab (notice 5ab%2B-7ab adds up to -2ab. So it is equivalent to -2ab)

5a%5E2%2Bhighlight%285ab%2B-7ab%29%2B-7b%5E2


Now let's factor 5a%5E2%2B5ab-7ab-7b%5E2 by grouping:


%285a%5E2%2B5ab%29%2B%28-7ab-7b%5E2%29 Group like terms


5a%28a%2Bb%29-7b%28a%2Bb%29 Factor out the GCF of 5a out of the first group. Factor out the GCF of -7b out of the second group


%285a-7b%29%28a%2Bb%29 Since we have a common term of a%2Bb, we can combine like terms

So 5a%5E2%2B5ab-7ab-7b%5E2 factors to %285a-7b%29%28a%2Bb%29


So this also means that 5a%5E2-2ab-7b%5E2 factors to %285a-7b%29%28a%2Bb%29 (since 5a%5E2-2ab-7b%5E2 is equivalent to 5a%5E2%2B5ab-7ab-7b%5E2)