SOLUTION: Are there two integers with a product of negative 16 and a sum of negative 4?

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Question 896901: Are there two integers with a product of negative 16 and a sum of negative 4?
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
x*y=-16,
x+y=-4
y=-x-4
x*(-x-4)=-16
-x^2-4x+16=0
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


-x%5E2-4%2Ax%2B16 Start with the given expression.



-%28x%5E2%2B4x-16%29 Factor out the GCF -1.



Now let's try to factor the inner expression x%5E2%2B4x-16



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Looking at the expression x%5E2%2B4x-16, we can see that the first coefficient is 1, the second coefficient is 4, and the last term is -16.



Now multiply the first coefficient 1 by the last term -16 to get %281%29%28-16%29=-16.



Now the question is: what two whole numbers multiply to -16 (the previous product) and add to the second coefficient 4?



To find these two numbers, we need to list all of the factors of -16 (the previous product).



Factors of -16:

1,2,4,8,16

-1,-2,-4,-8,-16



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



These factors pair up and multiply to -16.

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:



First NumberSecond NumberSum
1-161+(-16)=-15
2-82+(-8)=-6
4-44+(-4)=0
-116-1+16=15
-28-2+8=6
-44-4+4=0




From the table, we can see that there are no pairs of numbers which add to 4. So x%5E2%2B4x-16 cannot be factored.



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Answer:



So -x%5E2-4%2Ax%2B16 simply factors to -%28x%5E2%2B4x-16%29



In other words, -x%5E2-4%2Ax%2B16=-%28x%5E2%2B4x-16%29.