1. **State the problem:** Solve the quadratic equation $$6x^2 - 17x - 38 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 6$, $b = -17$, and $c = -38$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-17)^2 - 4 \times 6 \times (-38) = 289 + 912 = 1201$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-17) \pm \sqrt{1201}}{2 \times 6} = \frac{17 \pm \sqrt{1201}}{12}$$. 6. **Simplify the roots:** Since $\sqrt{1201}$ is irrational, the solutions are: $$x_1 = \frac{17 + \sqrt{1201}}{12}, \quad x_2 = \frac{17 - \sqrt{1201}}{12}$$. **Final answer:** $$x = \frac{17 \pm \sqrt{1201}}{12}$$.