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