k + k-1 = -2. k is a negative number satisfying this. What is the value of ?

Concept: For an equation of the form k + k-1 = c, multiplying both sides by k (k not equal to 0) clears the fraction into k2 - ck + 1 = 0. When c = -2, this becomes k2 + 2k + 1 = 0 - the perfect-square identity a2 + 2ab + b2 = (a + b)2 with a = k, b = 1 - so it factors as (k + 1)2 = 0. A repeated root like this pins down a single value of k, unlike a generic quadratic with two distinct roots.

Application:

1. Multiply the given equation by k: k2 + 1 = -2k.

2. Rearrange to standard form: k2 + 2k + 1 = 0.

3. Recognise the perfect-square identity: (k + 1)2 = 0, so k = -1 - which is indeed negative, matching the given condition.

4. Substitute k = -1 into the numerator: k2 + 4k - 2 = 1 - 4 - 2 = -5.

5. Substitute k = -1 into the denominator: k2 + k - 5 = 1 - 1 - 5 = -5.

6. Divide the two: (-5) / (-5) = 1.

Cross-check: Plugging k = -1 back into the original condition confirms it: -1 + (-1)-1 = -1 + (-1) = -2, which matches what was given. So the value of the expression is 1.