To solve the equation \( x^2 - 10x + 18 = x \) by factoring, we first move all terms to one side of the equation:
\[ x^2 - 10x + 18 - x = 0 \]
This simplifies to:
\[ x^2 - 11x + 18 = 0 \]
Next, we will factor the quadratic expression \( x^2 - 11x + 18 \).
We are looking for two numbers that multiply to \( 18 \) (the constant term) and add to \( -11 \) (the coefficient of \( x \)). The numbers \( -2 \) and \( -9 \) satisfy both conditions:
\[ -2 \cdot -9 = 18 \quad \text{and} \quad -2 + -9 = -11 \]
Thus, we can factor the quadratic as follows:
\[ (x - 2)(x - 9) = 0 \]
Setting each factor equal to zero gives us the possible solutions:
- \( x - 2 = 0 \) which leads to \( x = 2 \)
- \( x - 9 = 0 \) which leads to \( x = 9 \)
Therefore, the solutions to the equation \( x^2 - 10x + 18 = x \) are:
\[ \boxed{2 \text{ and } 9} \]