1. **State the problem:** Solve the quadratic equation $$3y^{2} - 24y - 47 = 0$$. 2. **Recall the quadratic formula:** For an equation $$ay^{2} + by + c = 0$$, the solutions are given by $$y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 3$$, $$b = -24$$, and $$c = -47$$. 4. **Calculate the discriminant:** $$\Delta = b^{2} - 4ac = (-24)^{2} - 4 \times 3 \times (-47) = 576 + 564 = 1140$$. 5. **Apply the quadratic formula:** $$y = \frac{-(-24) \pm \sqrt{1140}}{2 \times 3} = \frac{24 \pm \sqrt{1140}}{6}$$. 6. **Simplify the square root:** $$\sqrt{1140} = \sqrt{4 \times 285} = 2\sqrt{285}$$. 7. **Substitute back:** $$y = \frac{24 \pm 2\sqrt{285}}{6}$$. 8. **Simplify the fraction by canceling common factor 2:** $$y = \frac{\cancel{2} \times 12 \pm \cancel{2} \sqrt{285}}{\cancel{2} \times 3} = \frac{12 \pm \sqrt{285}}{3}$$. 9. **Final solutions:** $$y = \frac{12 + \sqrt{285}}{3} \quad \text{or} \quad y = \frac{12 - \sqrt{285}}{3}$$. These are the exact solutions to the quadratic equation.