The fastest solution most likely involves testing each answer choice. This allows us to solve the question in well under a minute.
That said, we can also solve the equation algebraically.

For equations involving square roots, we must make sure we test for extraneous roots (more in the video below).

Given: \(\sqrt{4 + x^{\frac{1}{2}}} =\sqrt{x + 2}\)

Square both sides to get: 4 + x^(1/2) = x + 2
Subtract 4 from both sides: x^(1/2) = x - 2
Square both sides: x = (x - 2)²
Expand: x = x² - 4x + 4
Subtract x from both sides: 0 = x² - 5x + 4
Factor: 0 = (x - 1)(x - 4)
So, EITHER x = 1 OR x = 4

IMPORTANT: Now plug the solutions into the original equation to check for extraneous roots.

Try x = 1
Given:\(\sqrt{4 + x^{\frac{1}{2}}} =\sqrt{x + 2}\)
Replace x with 1 to get: \(\sqrt{4 + 1^{\frac{1}{2}}} =\sqrt{1 + 2}\)
Simplify: \(\sqrt{5} =\sqrt{3}\)
Doesn't work.
ELIMINATE x = 1

Try x = 4
Given:\(\sqrt{4 + x^{\frac{1}{2}}} =\sqrt{x + 2}\)
Replace x with 4 to get: \(\sqrt{4 + 4^{\frac{1}{2}}} =\sqrt{4 + 2}\)
Simplify: \(\sqrt{4 + 2} =\sqrt{4 + 2}\)
Simplify: \(\sqrt{6} =\sqrt{6}\)
WORKS!!

Answer: D