1. **State the problem:** Solve the quadratic equation $$-2x^2 + 18x - 30 = 0$$. 2. **Write down the formula:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = -2$$, $$b = 18$$, and $$c = -30$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 18^2 - 4(-2)(-30) = 324 - 240 = 84$$. 5. **Apply the quadratic formula:** $$x = \frac{-18 \pm \sqrt{84}}{2(-2)} = \frac{-18 \pm \sqrt{84}}{-4}$$ 6. **Simplify the square root:** $$\sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}$$. 7. **Substitute back:** $$x = \frac{-18 \pm 2\sqrt{21}}{-4}$$ 8. **Simplify the fraction by canceling common factors:** $$x = \frac{\cancel{-18} \pm \cancel{2}\sqrt{21}}{\cancel{-4}} = \frac{-9 \pm \sqrt{21}}{-2}$$ 9. **Divide numerator and denominator by -1 to simplify signs:** $$x = \frac{9 \mp \sqrt{21}}{2}$$ 10. **Final solutions:** $$x_1 = \frac{9 + \sqrt{21}}{2}, \quad x_2 = \frac{9 - \sqrt{21}}{2}$$ These are the two solutions to the quadratic equation.