SOLUTION: a^2-7a-18

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Question 597660: a^2-7a-18
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor here.




Looking at the expression a%5E2-7a-18, we can see that the first coefficient is 1, the second coefficient is -7, and the last term is -18.


Now multiply the first coefficient 1 by the last term -18 to get %281%29%28-18%29=-18.


Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient -7?


To find these two numbers, we need to list all of the factors of -18 (the previous product).


Factors of -18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to -18.
1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -7:


First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3



From the table, we can see that the two numbers 2 and -9 add to -7 (the middle coefficient).


So the two numbers 2 and -9 both multiply to -18 and add to -7


Now replace the middle term -7a with 2a-9a. Remember, 2 and -9 add to -7. So this shows us that 2a-9a=-7a.


a%5E2%2Bhighlight%282a-9a%29-18 Replace the second term -7a with 2a-9a.


%28a%5E2%2B2a%29%2B%28-9a-18%29 Group the terms into two pairs.


a%28a%2B2%29%2B%28-9a-18%29 Factor out the GCF a from the first group.


a%28a%2B2%29-9%28a%2B2%29 Factor out 9 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28a-9%29%28a%2B2%29 Combine like terms. Or factor out the common term a%2B2


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Answer:


So a%5E2-7a-18 factors to %28a-9%29%28a%2B2%29.


In other words, a%5E2-7a-18=%28a-9%29%28a%2B2%29.


Note: you can check the answer by expanding %28a-9%29%28a%2B2%29 to get a%5E2-7a-18 or by graphing the original expression and the answer (the two graphs should be identical).