1. **State the problem:** Solve the quadratic equation $$8x^2 - 29x - 12 = 0$$. 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, $a = 8$, $b = -29$, and $c = -12$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-29)^2 - 4 \times 8 \times (-12) = 841 + 384 = 1225$$. 5. **Find the square root of the discriminant:** $$\sqrt{1225} = 35$$. 6. **Apply the quadratic formula:** $$x = \frac{-(-29) \pm 35}{2 \times 8} = \frac{29 \pm 35}{16}$$. 7. **Calculate the two solutions:** - $$x_1 = \frac{29 + 35}{16} = \frac{64}{16} = 4$$ - $$x_2 = \frac{29 - 35}{16} = \frac{-6}{16} = -\frac{3}{8}$$. **Final answer:** The solutions to the equation are $$x = 4$$ and $$x = -\frac{3}{8}$$.