Verified answer

Step-by-step explanation:

To solve for x when x + x^(-1) = -2, we can start by rearranging the equation:

x + 1/x = -2

Now, let's find a common denominator:

(x^2 + 1) / x = -2

Multiply both sides by x to get rid of the fraction:

x^2 + 1 = -2x

Now, rearrange the equation and set it equal to zero:

x^2 + 2x + 1 = 0

This is a quadratic equation, and we can factor it:

(x + 1)(x + 1) = 0

So, x = -1.

Now that we know the value of x, we can calculate x^2018 + 1/x^2019:

x^2018 + 1/x^2019 = (-1)^2018 + 1/(-1)^2019

Since any even power of -1 is 1, and any odd power of -1 is -1, we have:

x^2018 + 1/x^2019 = 1 + 1/(-1) = 1 - 1 = 0

So, x^2018 + 1/x^2019 is equal to 0.

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