1. **State the problem:** Solve the equation $$30 = 18x - 2x^2$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$0 = 18x - 2x^2 - 30$$ which can be rewritten as $$-2x^2 + 18x - 30 = 0$$ 3. **Simplify the equation:** Divide the entire equation by $-2$ to simplify coefficients: $$0 = \cancel{-2}x^2 - \cancel{18}x + \cancel{30} \div -2$$ which gives $$x^2 - 9x + 15 = 0$$ 4. **Use the quadratic formula:** For an equation $ax^2 + bx + c = 0$, the solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=1$, $b=-9$, and $c=15$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-9)^2 - 4(1)(15) = 81 - 60 = 21$$ 6. **Find the roots:** $$x = \frac{-(-9) \pm \sqrt{21}}{2(1)} = \frac{9 \pm \sqrt{21}}{2}$$ 7. **Final answer:** $$x = \frac{9 + \sqrt{21}}{2} \quad \text{or} \quad x = \frac{9 - \sqrt{21}}{2}$$ These are the two solutions to the equation.