To solve the quadratic equation \( x^2 - 6x + 8 = 0 \), we can use the factoring method. First, we look for two numbers that multiply to the constant term (8) and add up to the coefficient of the linear term (-6).

The numbers we're looking for are -2 and -4, since:

\[
-2 \times -4 = 8
\]
\[
-2 + (-4) = -6
\]

Now, we can rewrite the equation as:

\[
(x - 2)(x - 4) = 0
\]

Setting each factor equal to zero gives us:

1. \( x - 2 = 0 \) ⟹ \( x = 2 \)
2. \( x - 4 = 0 \) ⟹ \( x = 4 \)

Thus, the solutions to the equation \( x^2 - 6x + 8 = 0 \) are:

\[
x = 2 \quad \text{and} \quad x = 4
\]