SOLUTION: Factor completely: 8p^2 + 7pq - 18q^2 Can someone please explain to me in detail how to answer this question..

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Question 318368: Factor completely:
8p^2 + 7pq - 18q^2
Can someone please explain to me in detail how to answer this question..
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at 8p%5E2%2B7pq-18q%5E2 we can see that the first term is 8p%5E2 and the last term is -18q%5E2 where the coefficients are 8 and -18 respectively.

Now multiply the first coefficient 8 and the last coefficient -18 to get -144. Now what two numbers multiply to -144 and add to the middle coefficient 7? Let's list all of the factors of -144:



Factors of -144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -144
(1)*(-144)
(2)*(-72)
(3)*(-48)
(4)*(-36)
(6)*(-24)
(8)*(-18)
(9)*(-16)
(-1)*(144)
(-2)*(72)
(-3)*(48)
(-4)*(36)
(-6)*(24)
(-8)*(18)
(-9)*(16)

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


Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7

First NumberSecond NumberSum
1-1441+(-144)=-143
2-722+(-72)=-70
3-483+(-48)=-45
4-364+(-36)=-32
6-246+(-24)=-18
8-188+(-18)=-10
9-169+(-16)=-7
-1144-1+144=143
-272-2+72=70
-348-3+48=45
-436-4+36=32
-624-6+24=18
-818-8+18=10
-916-9+16=7



From this list we can see that -9 and 16 add up to 7 and multiply to -144


Now looking at the expression 8p%5E2%2B7pq-18q%5E2, replace 7pq with -9pq%2B16pq (notice -9pq%2B16pq adds up to 7pq. So it is equivalent to 7pq)

8p%5E2%2Bhighlight%28-9pq%2B16pq%29%2B-18q%5E2


Now let's factor 8p%5E2-9pq%2B16pq-18q%5E2 by grouping:


%288p%5E2-9pq%29%2B%2816pq-18q%5E2%29 Group like terms


p%288p-9q%29%2B2q%288p-9q%29 Factor out the GCF of p out of the first group. Factor out the GCF of 2q out of the second group


%28p%2B2q%29%288p-9q%29 Since we have a common term of 8p-9q, we can combine like terms


So 8p%5E2-9pq%2B16pq-18q%5E2 factors to %28p%2B2q%29%288p-9q%29


So this also means that 8p%5E2%2B7pq-18q%5E2 factors to %28p%2B2q%29%288p-9q%29 (since 8p%5E2%2B7pq-18q%5E2 is equivalent to 8p%5E2-9pq%2B16pq-18q%5E2)



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Answer:
So 8p%5E2%2B7pq-18q%5E2 factors to %28p%2B2q%29%288p-9q%29