Difference of squares: Using roots

To solve the equation $(\sqrt{x} + \frac{1}{2})(\sqrt{x} - \frac{1}{2}) = 0$, we can apply the zero-product property, which tells us that if a product of two factors is zero, at least one of the factors must be zero.

Let us proceed with each factor:

• First Factor: $\sqrt{x} + \frac{1}{2} = 0$
  Solving for $x$, subtract $\frac{1}{2}$ from both sides:
  $\sqrt{x} = -\frac{1}{2}$
  Squaring both sides, we get:
  $x = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}$.
  However, since the square root should be zero or positive, this case does not yield a real solution.
• Second Factor: $\sqrt{x} - \frac{1}{2} = 0$
  Solving for $x$, add $\frac{1}{2}$ to both sides:
  $\sqrt{x} = \frac{1}{2}$
  Squaring both sides, we have:
  $x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$.

Therefore, the solution to the equation $(\sqrt{x} + \frac{1}{2})(\sqrt{x} - \frac{1}{2}) = 0$ is $x = \frac{1}{4}$.

Upon reviewing the provided choices, the correct answer that matches our solution is: $\frac{1}{4}$ (Option 2).