1. **State the problem:** Solve the equation $$y^{2} + 48y - 64 = 0$$ for $y$. 2. **Identify the type of equation:** This is a quadratic equation in standard form $$ay^{2} + by + c = 0$$ where $a=1$, $b=48$, and $c=-64$. 3. **Recall the quadratic formula:** The solutions for $y$ are given by $$y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$ 4. **Calculate the discriminant:** $$\Delta = b^{2} - 4ac = 48^{2} - 4 \times 1 \times (-64) = 2304 + 256 = 2560$$ 5. **Apply the quadratic formula:** $$y = \frac{-48 \pm \sqrt{2560}}{2}$$ 6. **Simplify the square root:** $$\sqrt{2560} = \sqrt{256 \times 10} = 16\sqrt{10}$$ 7. **Substitute back:** $$y = \frac{-48 \pm 16\sqrt{10}}{2}$$ 8. **Simplify the fraction:** $$y = \frac{\cancel{-48}}{\cancel{2}} \pm \frac{16\sqrt{10}}{2} = -24 \pm 8\sqrt{10}$$ 9. **Final solutions:** $$y = -24 + 8\sqrt{10} \quad \text{or} \quad y = -24 - 8\sqrt{10}$$