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Let

f(n) = n^2 - 2 if n < 0

f(n) = 2n - 28 if n >= 0

 

What is the positive difference between the two values of a that satisfy the equation f(-2) + f(2) + f(a) = 0?

 Nov 11, 2021
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first, we should calculate the values of f(-2) and f(2) to be able to simplify the equation.

 

-2 is less than 0, so we're using the first function.

 

f(-2) = (-2)^2 - 2 = 4 - 2 = 2.

 

2 is greater than 0, so we're using the second function.

 

f(2) = 2(2) - 28 = 4 - 28 = -24

 

now let's plug it back into the equation and solve for a.

 

2 - 24 + f(a) = 0.

 

-22 + f(a) = 0

 

f(a) = 22.

 

since we now know what f(a) can be, we can work backwards using the two given functions.

 

let's start with the first function.

 

n^2 - 2 = 22

n^2 = 24

n = -sqrt(24) , as it's the only value less than 0 that satisfies the equation. (THOUGH: i'm not completely sure about this...)

 

second function,

 

2n - 28 = 22

2n = 50 

n = 25

 

so we get our difference to be 25 - (-sqrt(24)) which is 25 + sqrt24, or 25 + 2sqrt6.

 

i'm not completely sure about this answer though, hopefully it still helped you get some general idea of the problem - please let me know if you have any follow-up questions! smiley

 Nov 11, 2021

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