1. **State the problem:** Solve the quadratic equation $$-\frac{\sqrt{7}}{2}x^2 - 7x - \sqrt{7} = 0$$. 2. **Identify coefficients:** The quadratic equation is in the form $$ax^2 + bx + c = 0$$ where: $$a = -\frac{\sqrt{7}}{2}, \quad b = -7, \quad c = -\sqrt{7}$$. 3. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. **Calculate the discriminant:** $$b^2 - 4ac = (-7)^2 - 4 \times \left(-\frac{\sqrt{7}}{2}\right) \times (-\sqrt{7}) = 49 - 4 \times \left(-\frac{\sqrt{7}}{2}\right) \times (-\sqrt{7})$$ 5. Simplify the product inside the discriminant: $$4 \times \left(-\frac{\sqrt{7}}{2}\right) \times (-\sqrt{7}) = 4 \times \frac{\sqrt{7}}{2} \times \sqrt{7} = 4 \times \frac{7}{2} = 14$$ 6. Substitute back: $$b^2 - 4ac = 49 - 14 = 35$$ 7. **Calculate the roots:** $$x = \frac{-(-7) \pm \sqrt{35}}{2 \times \left(-\frac{\sqrt{7}}{2}\right)} = \frac{7 \pm \sqrt{35}}{-\sqrt{7}}$$ 8. Simplify the denominator: $$2 \times \left(-\frac{\sqrt{7}}{2}\right) = -\sqrt{7}$$ 9. Rewrite the expression: $$x = \frac{7 \pm \sqrt{35}}{-\sqrt{7}} = -\frac{7 \pm \sqrt{35}}{\sqrt{7}}$$ 10. Rationalize the denominator: $$x = -\frac{7 \pm \sqrt{35}}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = -\frac{7\sqrt{7} \pm \sqrt{35} \sqrt{7}}{7}$$ 11. Simplify the numerator: $$\sqrt{35} \sqrt{7} = \sqrt{35 \times 7} = \sqrt{245} = 7\sqrt{5}$$ 12. So, $$x = -\frac{7\sqrt{7} \pm 7\sqrt{5}}{7} = -\left(\sqrt{7} \pm \sqrt{5}\right)$$ 13. **Final solutions:** $$x = -\sqrt{7} - \sqrt{5} \quad \text{or} \quad x = -\sqrt{7} + \sqrt{5}$$