Answer:
The roots are real and distinct.
Step-by-step explanation:
Given the following equation:
[tex]x^{2} + kx - k -2 =0[/tex]
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,
[tex](-2)^{2} + 4(-2) + 8 = 4 - 8 + 8 = 4 [/tex]
At k = 0,
[tex](0)^2 + 4(0) + 8 = 8 [/tex]
At k = -100,
[tex](-100)^2 + 4(-100) + 8 = 10,000 - 400 + 8 = 9608[/tex]
Therefore, we can conclude that for all values of k, the roots are real and distinct.
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