SOLUTION: n^2-10n-24

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Question 613736: n^2-10n-24
Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression n%5E2-10n-24, we can see that the first coefficient is 1, the second coefficient is -10, and the last term is -24.


Now multiply the first coefficient 1 by the last term -24 to get %281%29%28-24%29=-24.


Now the question is: what two whole numbers multiply to -24 (the previous product) and add to the second coefficient -10?


To find these two numbers, we need to list all of the factors of -24 (the previous product).


Factors of -24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


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


These factors pair up and multiply to -24.
1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -10:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+6=2



From the table, we can see that the two numbers 2 and -12 add to -10 (the middle coefficient).


So the two numbers 2 and -12 both multiply to -24 and add to -10


Now replace the middle term -10n with 2n-12n. Remember, 2 and -12 add to -10. So this shows us that 2n-12n=-10n.


n%5E2%2Bhighlight%282n-12n%29-24 Replace the second term -10n with 2n-12n.


%28n%5E2%2B2n%29%2B%28-12n-24%29 Group the terms into two pairs.


n%28n%2B2%29%2B%28-12n-24%29 Factor out the GCF n from the first group.


n%28n%2B2%29-12%28n%2B2%29 Factor out 12 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.


%28n-12%29%28n%2B2%29 Combine like terms. Or factor out the common term n%2B2


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Answer:


So n%5E2-10n-24 factors to %28n-12%29%28n%2B2%29.


In other words, n%5E2-10n-24=%28n-12%29%28n%2B2%29.


Note: you can check the answer by expanding %28n-12%29%28n%2B2%29 to get n%5E2-10n-24 or by graphing the original expression and the answer (the two graphs should be identical).

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