The given quadratic equation is:

$x^2 - 2x - 15 = 0$

We need to solve this equation using the method of completing the square. Let's go through the steps:

Step 1: Move the constant to the other side

We start by moving the constant term $-15$ to the right-hand side of the equation:

$x^2 - 2x = 15$

Step 2: Complete the square

Next, we need to complete the square on the left-hand side. To do this, take half of the coefficient of $x$ (which is $-2$), square it, and add it to both sides.

$\text{Half of } -2 = -1$ $\text{Square of } -1 = (-1)^2 = 1$

Now, add 1 to both sides of the equation:

$x^2 - 2x + 1 = 15 + 1$ $x^2 - 2x + 1 = 16$

Step 3: Express the left side as a perfect square

The left-hand side is now a perfect square trinomial:

$(x - 1)^2 = 16$

Step 4: Solve for $x$

Take the square root of both sides:

$x - 1 = \pm \sqrt{16}$ $x - 1 = \pm 4$

Now, solve for $x$ by isolating it:

$x = 1 \pm 4$

This gives us two solutions:

1. $x = 1 + 4 = 5$
2. $x = 1 - 4 = -3$

Final Answer:

The solutions to the quadratic equation are:

$x = 5 \quad \text{or} \quad x = -3$

Would you like further details or explanations?

Here are 5 related questions to expand on this method:

1. How do you solve a quadratic equation using the quadratic formula?
2. What are the advantages of completing the square compared to factoring?
3. How is the method of completing the square useful in deriving the quadratic formula?
4. Can completing the square be applied to any quadratic equation?
5. How does completing the square help in graphing quadratic functions?

Tip: Completing the square is particularly useful when you want to convert a quadratic equation into vertex form, which is helpful for analyzing the graph of the function.