1. **State the problem:** Solve the quadratic equation $$7x^2 - 2x - 8 = 0$$. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 7$$, $$b = -2$$, and $$c = -8$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-2)^2 - 4 \times 7 \times (-8) = 4 + 224 = 228$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-2) \pm \sqrt{228}}{2 \times 7} = \frac{2 \pm \sqrt{228}}{14}$$. 6. **Simplify the square root:** $$\sqrt{228} = \sqrt{4 \times 57} = 2\sqrt{57}$$. 7. **Substitute back:** $$x = \frac{2 \pm 2\sqrt{57}}{14}$$. 8. **Cancel common factor 2:** $$x = \frac{\cancel{2} \pm \cancel{2}\sqrt{57}}{\cancel{14}} = \frac{1 \pm \sqrt{57}}{7}$$. 9. **Final solutions:** $$x_1 = \frac{1 + \sqrt{57}}{7}, \quad x_2 = \frac{1 - \sqrt{57}}{7}$$. These are the exact solutions to the quadratic equation.