Choice C is the correct answer. To solve for $x$, you must isolate the variable term $x^2$. Following the reverse order of operations (SADMEP):

1. Subtract/Add: Subtract 5 from both sides ($2x^2 = 18$).
2. Divide/Multiply: Divide by 2 ($x^2 = 9$).
3. Exponents: Take the square root ($x = \pm 3$).

Choice A is incorrect because dividing by 2 first would create fractions ($x^2 + 2.5 = 11.5$), which is valid but messier. Choice B is incorrect because you cannot take the square root of sums individually. Choice D is incorrect because there is no x term to factor.