Question

Show that the roots of the equations

Answer 1

Answer:

The roots are real and distinct.

Step-by-step explanation:

Given the following equation:

$x^{2} + kx - k -2 =0$

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,

$(-2)^{2} + 4(-2) + 8 = 4 - 8 + 8 = 4$

At k = 0,

$(0)^2 + 4(0) + 8 = 8$

At k = -100,

$(-100)^2 + 4(-100) + 8 = 10,000 - 400 + 8 = 9608$

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

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