What is the solution for x in the equation sqrt(x + 2) - sqrt(x) = 1?

To solve the equation (\sqrt{x + 2} - \sqrt{x} = 1), the first step is to isolate one of the square roots. We can rewrite the equation as:

(\sqrt{x + 2} = \sqrt{x} + 1).

Next, we square both sides to eliminate the square roots:

((\sqrt{x + 2})^2 = (\sqrt{x} + 1)^2).

This results in:

(x + 2 = x + 2\sqrt{x} + 1).

Now, we can simplify this to:

(x + 2 = x + 2\sqrt{x} + 1).

Subtracting (x) from both sides gives:

(2 = 2\sqrt{x} + 1).

Next, isolate (2\sqrt{x}):

(2\sqrt{x} = 2 - 1),

(2\sqrt{x} = 1).

Now, divide both sides by 2:

(\sqrt{x} = \frac{1}{2}).

Finally, we square both sides again to solve for (x):

((\sqrt{x})^2 =