1. **State the problem:** Solve the equation $$\sqrt{3} \over 130 \cdot v^2 + \frac{4488 - 2176\sqrt{3}}{13} v + \frac{115600}{65} = 0$$ for $v$. 2. **Identify the type of equation:** This is a quadratic equation in the form $$a v^2 + b v + c = 0$$ where: $$a = \frac{\sqrt{3}}{130}, \quad b = \frac{4488 - 2176\sqrt{3}}{13}, \quad c = \frac{115600}{65}$$ 3. **Recall the quadratic formula:** For $a v^2 + b v + c = 0$, the solutions are: $$v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Calculate the discriminant $\Delta = b^2 - 4ac$:** $$b^2 = \left(\frac{4488 - 2176\sqrt{3}}{13}\right)^2$$ $$4ac = 4 \times \frac{\sqrt{3}}{130} \times \frac{115600}{65}$$ Calculate each term: $$b^2 = \frac{(4488 - 2176\sqrt{3})^2}{169}$$ $$4ac = \frac{4 \times \sqrt{3} \times 115600}{130 \times 65} = \frac{4 \times \sqrt{3} \times 115600}{8450}$$ Simplify numerator and denominator: $$4ac = \frac{462400 \sqrt{3}}{8450} = \frac{462400 \sqrt{3}}{8450}$$ 5. **Expand $(4488 - 2176\sqrt{3})^2$:** Use $(x - y)^2 = x^2 - 2xy + y^2$: $$4488^2 - 2 \times 4488 \times 2176 \sqrt{3} + (2176 \sqrt{3})^2$$ Calculate each term: $$4488^2 = 20142244$$ $$2 \times 4488 \times 2176 = 19543872$$ $$ (2176 \sqrt{3})^2 = 2176^2 \times 3 = 4734976 \times 3 = 14204928$$ So: $$b^2 = \frac{20142244 - 19543872 \sqrt{3} + 14204928}{169} = \frac{34347172 - 19543872 \sqrt{3}}{169}$$ 6. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = \frac{34347172 - 19543872 \sqrt{3}}{169} - \frac{462400 \sqrt{3}}{8450}$$ Find common denominator $169 \times 50 = 8450$: $$\Delta = \frac{(34347172 - 19543872 \sqrt{3}) \times 50}{8450} - \frac{462400 \sqrt{3}}{8450}$$ $$= \frac{171735860 - 977193600 \sqrt{3} - 462400 \sqrt{3}}{8450} = \frac{171735860 - 977656000 \sqrt{3}}{8450}$$ 7. **Calculate $v$ using quadratic formula:** $$v = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-\frac{4488 - 2176 \sqrt{3}}{13} \pm \sqrt{\frac{171735860 - 977656000 \sqrt{3}}{8450}}}{2 \times \frac{\sqrt{3}}{130}}$$ Simplify denominator: $$2a = \frac{2 \sqrt{3}}{130} = \frac{\sqrt{3}}{65}$$ Rewrite numerator: $$-b = -\frac{4488 - 2176 \sqrt{3}}{13} = \frac{-4488 + 2176 \sqrt{3}}{13}$$ 8. **Simplify the fraction by multiplying numerator and denominator by 65:** $$v = \frac{\frac{-4488 + 2176 \sqrt{3}}{13} \pm \sqrt{\frac{171735860 - 977656000 \sqrt{3}}{8450}}}{\frac{\sqrt{3}}{65}} = \frac{65}{\sqrt{3}} \left( \frac{-4488 + 2176 \sqrt{3}}{13} \pm \sqrt{\frac{171735860 - 977656000 \sqrt{3}}{8450}} \right)$$ 9. **Final answer:** $$v = \frac{65}{\sqrt{3}} \left( \frac{-4488 + 2176 \sqrt{3}}{13} \pm \sqrt{\frac{171735860 - 977656000 \sqrt{3}}{8450}} \right)$$ This expression gives the two possible values of $v$ that satisfy the original quadratic equation.