1. **State the problem:** Solve the quadratic equation $$\frac{\sqrt{3}}{130}v^2 - \frac{4488 - 2176\sqrt{3}}{13}v + \frac{115600}{65} = 0$$ for $v$, giving all answers to three decimal places. 2. **Formula used:** For a quadratic equation $$ax^2 + bx + c = 0$$, the solutions are given by the quadratic formula: $$v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 3. **Identify coefficients:** $$a = \frac{\sqrt{3}}{130}, \quad b = -\frac{4488 - 2176\sqrt{3}}{13}, \quad c = \frac{115600}{65}$$ 4. **Calculate discriminant:** $$\Delta = b^2 - 4ac$$ 5. **Calculate $b^2$:** $$b^2 = \left(-\frac{4488 - 2176\sqrt{3}}{13}\right)^2 = \frac{(4488 - 2176\sqrt{3})^2}{169}$$ 6. **Calculate $4ac$:** $$4ac = 4 \times \frac{\sqrt{3}}{130} \times \frac{115600}{65} = \frac{4 \times \sqrt{3} \times 115600}{130 \times 65}$$ 7. **Simplify $4ac$:** $$130 \times 65 = 8450$$ $$4ac = \frac{4 \times 115600 \times \sqrt{3}}{8450} = \frac{462400 \sqrt{3}}{8450}$$ 8. **Calculate discriminant numerically:** Calculate $b$ numerically: $$4488 - 2176\sqrt{3} \approx 4488 - 2176 \times 1.732 = 4488 - 3769.632 = 718.368$$ So, $$b = -\frac{718.368}{13} \approx -55.259$$ Calculate $a$ numerically: $$a = \frac{1.732}{130} \approx 0.01332$$ Calculate $c$ numerically: $$c = \frac{115600}{65} = 1778.462$$ Calculate discriminant: $$\Delta = b^2 - 4ac = (-55.259)^2 - 4 \times 0.01332 \times 1778.462$$ $$= 3053.56 - 94.82 = 2958.74$$ 9. **Calculate roots:** $$v = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-55.259) \pm \sqrt{2958.74}}{2 \times 0.01332} = \frac{55.259 \pm 54.39}{0.02664}$$ 10. **Calculate each root:** $$v_1 = \frac{55.259 + 54.39}{0.02664} = \frac{109.649}{0.02664} \approx 4115.785$$ $$v_2 = \frac{55.259 - 54.39}{0.02664} = \frac{0.869}{0.02664} \approx 32.609$$ **Final answers:** $$v \approx 4115.785, \quad v \approx 32.609$$