Solve the quadratic equation by completing the square: x^2 - 2x - 3 = 0.

A. x = -1, 3

B. x = 1, -3

C. x = 3, -1

D. x = -3, 1

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.

Solve the quadratic equation by completing the square: x^2 - 2x - 3 = 0.

Learn how to solve the quadratic equation x^2 - 2x - 3 = 0 by completing the square. This step-by-step guide covers key algebraic concepts and techniques, including completing the square and solving for x. Suitable for students in Grades 8-10.

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