SOLUTION: Hello, thank you much for taking your time to help me! It's much appreciated. I need help factoring an equation. I know the original equation and I know it's factored answer.

Algebra ->  Equations -> SOLUTION: Hello, thank you much for taking your time to help me! It's much appreciated. I need help factoring an equation. I know the original equation and I know it's factored answer.      Log On


   

Question 407874: Hello, thank you much for taking your time to help me! It's much appreciated.
I need help factoring an equation.
I know the original equation and I know it's factored answer.
The part that I can't understand is how to get there.
The equation is: 6a^2 - 9ab - 15b^2
The answer is: 3(2a - 5b)(a + b)
When I factor it, I reach: 3(2a^2 - 3ab - 5b^2)
Where do I go from here? How do I do it? Why do I do it that way?
Any help is beyond loved!!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

6a%5E2-9ab-15b%5E2 Start with the given expression


3%282a%5E2-3ab-5b%5E2%29 Factor out the GCF 3


Now let's focus on the inner expression 2a%5E2-3ab-5b%5E2




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Looking at 2a%5E2-3ab-5b%5E2 we can see that the first term is 2a%5E2 and the last term is -5b%5E2 where the coefficients are 2 and -5 respectively.

Now multiply the first coefficient 2 and the last coefficient -5 to get -10. Now what two numbers multiply to -10 and add to the middle coefficient -3? Let's list all of the factors of -10:



Factors of -10:
1,2,5,10

-1,-2,-5,-10 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -10
(1)*(-10)
(2)*(-5)
(-1)*(10)
(-2)*(5)

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


Now which of these pairs add to -3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -3

First NumberSecond NumberSum
1-101+(-10)=-9
2-52+(-5)=-3
-110-1+10=9
-25-2+5=3



From this list we can see that 2 and -5 add up to -3 and multiply to -10


Now looking at the expression 2a%5E2-3ab-5b%5E2, replace -3ab with 2ab%2B-5ab (notice 2ab%2B-5ab adds up to -3ab. So it is equivalent to -3ab)

2a%5E2%2Bhighlight%282ab%2B-5ab%29%2B-5b%5E2


Now let's factor 2a%5E2%2B2ab-5ab-5b%5E2 by grouping:


%282a%5E2%2B2ab%29%2B%28-5ab-5b%5E2%29 Group like terms


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


%282a-5b%29%28a%2Bb%29 Since we have a common term of a%2Bb, we can combine like terms

So 2a%5E2%2B2ab-5ab-5b%5E2 factors to %282a-5b%29%28a%2Bb%29


So this also means that 2a%5E2-3ab-5b%5E2 factors to %282a-5b%29%28a%2Bb%29 (since 2a%5E2-3ab-5b%5E2 is equivalent to 2a%5E2%2B2ab-5ab-5b%5E2)



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So our expression goes from 3%282a%5E2-3ab-5b%5E2%29 and factors further to 3%282a-5b%29%28a%2Bb%29


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Answer:

So 6a%5E2-9ab-15b%5E2 factors to 3%282a-5b%29%28a%2Bb%29

If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim