We are given the quadratic equation:

$x^2 - 2x - 3 = 0$

To solve by completing the square, follow these steps:

Step 1: Move the constant to the other side of the equation.

$x^2 - 2x = 3$

Step 2: Complete the square.

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides.

The coefficient of $x$ is $-2$, so: $\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1$

Add 1 to both sides: $x^2 - 2x + 1 = 3 + 1$ $(x - 1)^2 = 4$

Step 3: Solve for $x$.

Take the square root of both sides: $x - 1 = \pm 2$

Step 4: Solve for $x$ by isolating it.

$x - 1 = 2 \quad \text{or} \quad x - 1 = -2$ $x = 3 \quad \text{or} \quad x = -1$

Thus, the solutions are $x = 3$ and $x = -1$, which corresponds to option:

C. $x = 3, -1$

Would you like more details on any step? Here are five related questions to explore further:

1. How does completing the square relate to the quadratic formula?
2. Why do we take half of the coefficient of $x$ in completing the square?
3. What is the geometric interpretation of completing the square?
4. Can you complete the square for the equation $x^2 + 4x - 5 = 0$?
5. What are other methods to solve quadratic equations?

Tip: Completing the square is useful not only for solving equations but also for converting quadratics to vertex form.