1. **State the problem:** Solve the quadratic equation $$64x^2 + 5x + 25 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 64$$, $$b = 5$$, and $$c = 25$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 64 \times 25 = 25 - 6400 = -6375$$. 5. Since $$\Delta < 0$$, the equation has no real roots but two complex roots. 6. **Apply the quadratic formula:** $$x = \frac{-5 \pm \sqrt{-6375}}{2 \times 64} = \frac{-5 \pm i\sqrt{6375}}{128}$$. 7. **Simplify the square root:** $$6375 = 25 \times 255 = 25 \times 15 \times 17$$ $$\sqrt{6375} = \sqrt{25 \times 255} = 5\sqrt{255}$$. 8. **Final solution:** $$x = \frac{-5 \pm 5i\sqrt{255}}{128} = \frac{-5}{128} \pm \frac{5i\sqrt{255}}{128}$$. Thus, the two complex solutions are $$x = \frac{-5}{128} + \frac{5i\sqrt{255}}{128}$$ and $$x = \frac{-5}{128} - \frac{5i\sqrt{255}}{128}$$.