To solve the equation \( x^2 - 10x + 21 = x - 3 \) by factoring, we first rearrange the equation to set it to zero:
\[ x^2 - 10x + 21 - x + 3 = 0 \]
Combine like terms:
\[ x^2 - 11x + 24 = 0 \]
Next, we need to factor the quadratic expression \( x^2 - 11x + 24 \). We look for two numbers that multiply to \( 24 \) (the constant term) and add to \( -11 \) (the coefficient of \( x \)). These numbers are \( -3 \) and \( -8 \). Therefore, we can factor the expression as follows:
\[ (x - 3)(x - 8) = 0 \]
Now, we can set each factor equal to zero:
-
\( x - 3 = 0 \)
- This gives \( x = 3 \).
-
\( x - 8 = 0 \)
- This gives \( x = 8 \).
Thus, the solutions to the equation \( x^2 - 10x + 21 = x - 3 \) are:
\[ \boxed{3} \quad \text{and} \quad \boxed{8} \]