Show that the roots of the equations x^2 + kx - k - 2 = 0 are real and distinct for all
values of k.

The roots are real and distinct.

Step-by-step explanation:

Given the following equation:

In this problem, a = 1, b = k and c = -k - 2

The discriminant is b² - 4ac, and for the roots to be real and distinct, it must be at least or greater than 0.

We get,

(k)²- 4(1)(-k - 2) = 1 - 4(-k - 2)

= k² + 4k + 8

Let's check:

At k = -2,

At k = 0,

At k = -100,

Therefore, we can conclude that for all values of k, the roots are real and distinct.

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