1. **State the problem:** Factor the quadratic expression $x^2 - 24x + 63$ into the form $(x - a)(x - b)$. 2. **Recall the factoring formula:** For a quadratic $x^2 + bx + c$, the factors are $(x - m)(x - n)$ where $m$ and $n$ satisfy $m + n = -b$ and $mn = c$. 3. **Apply the formula:** Here, $b = -24$ and $c = 63$. So, we need two numbers $m$ and $n$ such that $$m + n = 24$$ $$mn = 63$$ 4. **Find the numbers:** Factors of 63 are 1 and 63, 3 and 21, 7 and 9. Among these, 7 and 9 add up to 16, 3 and 21 add up to 24. So, $m = 3$ and $n = 21$. 5. **Write the factorization:** Therefore, $$x^2 - 24x + 63 = (x - 3)(x - 21)$$ 6. **Check:** Expanding $(x - 3)(x - 21)$ gives $x^2 - 21x - 3x + 63 = x^2 - 24x + 63$, which matches the original expression. **Final answer:** $$\boxed{(x - 3)(x - 21)}$$