To solve the equation \( x^2 - 9x + 8 = 0 \) by factoring, we need to factor the quadratic expression on the left side of the equation.
Step 1: Look for two numbers that multiply to give the constant term (8) and add up to give the coefficient of the linear term (-9).
The two numbers that meet these criteria are -8 and -1, because:
\[ -8 \times -1 = 8 \] \[ -8 + (-1) = -9 \]
Step 2: Now we can factor the quadratic:
\[ x^2 - 9x + 8 = (x - 8)(x - 1) \]
Step 3: Set the factored equation equal to zero:
\[ (x - 8)(x - 1) = 0 \]
Step 4: Solve for \( x \) by setting each factor equal to zero:
- \( x - 8 = 0 \) gives \( x = 8 \)
- \( x - 1 = 0 \) gives \( x = 1 \)
Thus, the solutions to the equation \( x^2 - 9x + 8 = 0 \) are:
\[ \boxed{8} \quad \text{and} \quad \boxed{1} \]