SOLUTION: I am needing some help with this... Show work, Factor and if Prime state this... 10a^2+25a-15

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Question 405580: I am needing some help with this...
Show work, Factor and if Prime state this...
10a^2+25a-15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

10a%5E2%2B25a-15 Start with the given expression


5%282a%5E2%2B5a-3%29 Factor out the GCF 5


Now let's focus on the inner expression 2a%5E2%2B5a-3




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Looking at 2a%5E2%2B5a-3 we can see that the first term is 2a%5E2 and the last term is -3 where the coefficients are 2 and -3 respectively.

Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the middle coefficient 5? Let's list all of the factors of -6:



Factors of -6:
1,2,3,6

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

These factors pair up and multiply to -6
(1)*(-6)
(2)*(-3)
(-1)*(6)
(-2)*(3)

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
1-61+(-6)=-5
2-32+(-3)=-1
-16-1+6=5
-23-2+3=1



From this list we can see that -1 and 6 add up to 5 and multiply to -6


Now looking at the expression 2a%5E2%2B5a-3, replace 5a with -a%2B6a (notice -a%2B6a adds up to 5a. So it is equivalent to 5a)

2a%5E2%2Bhighlight%28-a%2B6a%29-3


Now let's factor 2a%5E2-a%2B6a-3 by grouping:


%282a%5E2-a%29%2B%286a-3%29 Group like terms


a%282a-1%29%2B3%282a-1%29 Factor out the GCF of a out of the first group. Factor out the GCF of 3 out of the second group


%28a%2B3%29%282a-1%29 Since we have a common term of 2a-1, we can combine like terms

So 2a%5E2-a%2B6a-3 factors to %28a%2B3%29%282a-1%29


So this also means that 2a%5E2%2B5a-3 factors to %28a%2B3%29%282a-1%29 (since 2a%5E2%2B5a-3 is equivalent to 2a%5E2-a%2B6a-3)



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So our expression goes from 5%282a%5E2%2B5a-3%29 and factors further to 5%28a%2B3%29%282a-1%29


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Answer:

So 10a%5E2%2B25a-15 factors to 5%28a%2B3%29%282a-1%29

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

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Jim