1. **State the problem:** Solve the quadratic equation $$5x^2 - 13x - 6 = 0$$. 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, $a = 5$, $b = -13$, and $c = -6$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-13)^2 - 4 \times 5 \times (-6) = 169 + 120 = 289$$. 5. **Find the square root of the discriminant:** $$\sqrt{289} = 17$$. 6. **Apply the quadratic formula:** $$x = \frac{-(-13) \pm 17}{2 \times 5} = \frac{13 \pm 17}{10}$$. 7. **Calculate the two solutions:** - For the plus sign: $$x = \frac{13 + 17}{10} = \frac{30}{10} = 3$$. - For the minus sign: $$x = \frac{13 - 17}{10} = \frac{\cancel{13 - 17}}{10} = \frac{-4}{10} = -\frac{2}{5}$$. 8. **Final answer:** The solutions to the equation are $$x = 3$$ and $$x = -\frac{2}{5}$$.