To solve the quadratic equation $x^2 - 2x - 8 = 0$ by completing the square, follow these steps:

1. Rewrite the equation: $x^2 - 2x - 8 = 0$ Add 8 to both sides to isolate the quadratic and linear terms on one side: $x^2 - 2x = 8$

2. Complete the square: To complete the square, add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is $-2$, so half of this is $-1$, and squaring it gives $1$. Add and subtract $1$ to the left side: $x^2 - 2x + 1 - 1 = 8$ Simplify the left side: $(x - 1)^2 - 1 = 8$ Add 1 to both sides: $(x - 1)^2 = 9$

3. Solve for $x$: Take the square root of both sides: $x - 1 = \pm 3$ Solve for $x$ by adding 1 to both sides for each case: $x - 1 = 3 \quad \Rightarrow \quad x = 4$ $x - 1 = -3 \quad \Rightarrow \quad x = -2$

The solutions to the equation $x^2 - 2x - 8 = 0$ are $x = 4$ and $x = -2$.

Would you like more details or have any questions? Here are some related questions to expand your understanding:

1. How can you verify the solutions $x = 4$ and $x = -2$ in the original equation?
2. What is the purpose of completing the square when solving quadratic equations?
3. Can all quadratic equations be solved by completing the square?
4. How does the method of completing the square compare to using the quadratic formula?
5. What are the benefits and drawbacks of completing the square versus factoring for solving quadratics?

Tip: When completing the square, always ensure to balance the equation by adding and subtracting the same value, and keep the equation balanced throughout the process.