1. **State the problem:** Solve the quadratic equation $$6x - x(x - 13) = 18$$ for $x$. 2. **Rewrite the equation:** Expand the term $x(x - 13)$: $$6x - (x^2 - 13x) = 18$$ 3. **Simplify the equation:** Distribute the minus sign: $$6x - x^2 + 13x = 18$$ Combine like terms: $$-x^2 + 19x = 18$$ 4. **Bring all terms to one side to set the equation to zero:** $$-x^2 + 19x - 18 = 0$$ Multiply both sides by $-1$ to make the leading coefficient positive: $$\cancel{-1}(-x^2 + 19x - 18) = \cancel{-1}(0)$$ $$x^2 - 19x + 18 = 0$$ 5. **Use the quadratic formula:** The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-19$, and $c=18$. 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-19)^2 - 4(1)(18) = 361 - 72 = 289$$ 7. **Find the square root of the discriminant:** $$\sqrt{289} = 17$$ 8. **Calculate the two solutions:** $$x_1 = \frac{-(-19) + 17}{2(1)} = \frac{19 + 17}{2} = \frac{36}{2} = 18$$ $$x_2 = \frac{-(-19) - 17}{2(1)} = \frac{19 - 17}{2} = \frac{2}{2} = 1$$ **Final answer:** $$x_1 = 18, \quad x_2 = 1$$