For what values of a does equation (a-2) x = a ^ 2-4 have only one positive root?

1. For a = 2, the equation turns into a true equality with an infinite number of roots:

(a – 2) x = a ^ 2 – 4;
(2 – 2) x = 2 ^ 2 – 4;
0 = 0;
x ∈ (-∞; ∞).
2. For a ≠ 2, the equation has a single root:

(a – 2) x = (a + 2) (a – 2);
x = (a + 2) (a – 2) / (a – 2);
x = a + 2.
3. The root of the equation will be positive provided:

a + 2> 0;
a> -2;
a ∈ (-2; ∞).
4. Excluding the value a = 2 from this interval, we get:

a ∈ (-2; 2) ∪ (2; ∞).

Answer: a ∈ (-2; 2) ∪ (2; ∞).