1. **State the problem:** Solve the quadratic equation $x^2 - 18x + 81 = 0$. 2. **Recall the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-18$, and $c=81$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-18)^2 - 4 \times 1 \times 81 = 324 - 324 = 0$$ 4. **Interpret the discriminant:** Since $\Delta = 0$, there is exactly one real root (a repeated root). 5. **Find the root:** $$x = \frac{-(-18)}{2 \times 1} = \frac{18}{2} = 9$$ 6. **Verify by factoring:** $$x^2 - 18x + 81 = (x - 9)^2 = 0$$ which confirms the root $x=9$. **Final answer:** The solution to the equation is $x = 9$.