1. **State the problem:** Solve the quadratic equation $$2x^2 - 5x - 12 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 2$$, $$b = -5$$, and $$c = -12$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 2 \times (-12) = 25 + 96 = 121$$. 5. **Find the square root of the discriminant:** $$\sqrt{121} = 11$$. 6. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm 11}{2 \times 2} = \frac{5 \pm 11}{4}$$. 7. **Calculate the two solutions:** - $$x_1 = \frac{5 + 11}{4} = \frac{16}{4} = 4$$ - $$x_2 = \frac{5 - 11}{4} = \frac{-6}{4} = -\frac{3}{2}$$. 8. **Final answer:** The solutions to the equation $$2x^2 - 5x - 12 = 0$$ are $$x = 4$$ and $$x = -\frac{3}{2}$$.