To solve the quadratic equation $x^2 - 6x + 5 = 0$ by completing the square, follow these steps:

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

Start by rewriting the equation:

$x^2 - 6x = -5$

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides of the equation. The coefficient of $x$ is $-6$, so:

$\left(\frac{-6}{2}\right)^2 = 9$

Add 9 to both sides of the equation:

$x^2 - 6x + 9 = -5 + 9$

$(x - 3)^2 = 4$

Step 3: Solve for $x$

Now take the square root of both sides:

$x - 3 = \pm 2$

So,

$x = 3 \pm 2$

This gives two solutions:

$x = 3 + 2 = 5 \quad \text{or} \quad x = 3 - 2 = 1$

Final Answer:

The solutions are $x = 5$ and $x = 1$. Thus, the correct option is: $\boxed{x = 5, 1}$

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does completing the square compare to using the quadratic formula?
2. Can you apply the method of completing the square to other forms of quadratics, such as those with a leading coefficient other than 1?
3. How would the solution change if the quadratic term had a different sign (e.g., $-x^2$)?
4. What is the geometric interpretation of completing the square on a quadratic function?
5. How can completing the square help in deriving the quadratic formula?

Tip: When completing the square, always ensure that the coefficient of $x^2$ is 1 before proceeding. If it isn't, divide the entire equation by that coefficient first!