1. **State the problem:** Factor the quadratic expression $$x^2 - 24x = 63$$ into the form $$(x - a)(x - b)$$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$x^2 - 24x - 63 = 0$$ 3. **Identify coefficients:** Here, $a = 1$, $b = -24$, and $c = -63$. 4. **Use the factoring method:** We look for two numbers that multiply to $ac = 1 \times (-63) = -63$ and add to $b = -24$. 5. **Find the pair:** The numbers are $-27$ and $3$ because: $$-27 \times 3 = -81$$ (incorrect), so try again. Actually, check carefully: We need two numbers that multiply to $-63$ and add to $-24$. Try $-27$ and $3$: $$-27 + 3 = -24$$ $$-27 \times 3 = -81$$ (not $-63$) Try $-21$ and $3$: $$-21 + 3 = -18$$ (no) Try $-9$ and $7$: $$-9 + 7 = -2$$ (no) Try $-63$ and $1$: $$-63 + 1 = -62$$ (no) Try $-3$ and $21$: $$-3 + 21 = 18$$ (no) Try $-1$ and $63$: $$-1 + 63 = 62$$ (no) Try $-18$ and $3.5$ (not integer, discard) Since no integer pair works, use the quadratic formula: 6. **Quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{24 \pm \sqrt{(-24)^2 - 4 \times 1 \times (-63)}}{2}$$ Calculate discriminant: $$576 + 252 = 828$$ 7. **Simplify square root:** $$\sqrt{828} = \sqrt{4 \times 207} = 2\sqrt{207}$$ 8. **Final roots:** $$x = \frac{24 \pm 2\sqrt{207}}{2} = 12 \pm \sqrt{207}$$ 9. **Write factorization:** $$x^2 - 24x - 63 = (x - (12 + \sqrt{207}))(x - (12 - \sqrt{207}))$$ **Answer:** The factorization is $$(x - (12 + \sqrt{207}))(x - (12 - \sqrt{207}))$$.