1. **State the problems:** Solve the quadratic equation $9x^2 - 3x + 1 = 0$ and then prove that $$81\left(x^4 + \frac{1}{6561x^4}\right) = 1.$$ 2. **Solve the quadratic equation:** Given: $$9x^2 - 3x + 1 = 0.$$ Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$ where $a=9$, $b=-3$, $c=1$. Calculate the discriminant: $$\Delta = (-3)^2 - 4 \times 9 \times 1 = 9 - 36 = -27.$$ Since the discriminant is negative, the roots are complex: $$x = \frac{3 \pm \sqrt{-27}}{18} = \frac{3 \pm 3i\sqrt{3}}{18} = \frac{1 \pm i\sqrt{3}}{6}.$$ 3. **Calculate $x^4$:** Let $x = \frac{1 + i\sqrt{3}}{6}$ (the process will be similar for the conjugate). First find $6x = 1 + i\sqrt{3}$. Note that $1 + i\sqrt{3} = 2\left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) = 2 e^{i \pi/3}$. Then, $$ (6x)^4 = (2 e^{i \pi/3})^4 = 2^4 e^{i 4\pi/3} = 16 e^{i 4\pi/3}.$$ Consequently, $$ x^4 = \frac{(6x)^4}{6^4} = \frac{16 e^{i 4\pi/3}}{1296} = \frac{16}{1296} e^{i 4\pi/3} = \frac{1}{81} e^{i 4\pi/3}.$$ 4. **Calculate $\frac{1}{6561 x^4}$:** Since $6561 = 81^2$, we get $$ \frac{1}{6561 x^4} = \frac{1}{81^2 x^4} = \frac{1}{81^2} \cdot \frac{1}{x^4} = \frac{1}{81^2} \cdot \frac{1}{\frac{1}{81} e^{i 4\pi/3}} = \frac{1}{81} e^{-i 4\pi/3}.$$ 5. **Add $x^4$ and $\frac{1}{6561 x^4}$:** $$x^4 + \frac{1}{6561 x^4} = \frac{1}{81} e^{i 4\pi/3} + \frac{1}{81} e^{-i 4\pi/3} = \frac{1}{81} \left(e^{i 4\pi/3} + e^{-i 4\pi/3}\right).$$ Recall that $e^{i\theta} + e^{-i\theta} = 2 \cos \theta$, so: $$x^4 + \frac{1}{6561 x^4} = \frac{2}{81} \cos \left( \frac{4\pi}{3} \right).$$ 6. **Evaluate $\cos(4\pi/3)$:** $$\cos \left( \frac{4\pi}{3} \right) = \cos \left( \pi + \frac{\pi}{3} \right) = -\cos \left( \frac{\pi}{3} \right) = - \frac{1}{2}.$$ 7. **Plug the cosine value back:** $$x^4 + \frac{1}{6561 x^4} = \frac{2}{81} \times \left(- \frac{1}{2}\right) = - \frac{1}{81}.$$ 8. **Multiply both sides by 81:** $$81 \left(x^4 + \frac{1}{6561 x^4} \right) = 81 \times \left(- \frac{1}{81} \right) = -1.$$ **However, the problem states to prove this equals 1, but our calculation gives -1.** 9. **Explanation:** The value is $-1$ given the roots and calculations above. Possibly the problem expects a slightly different expression or there might be a typo. **Final solution:** The quadratic solutions are $$x = \frac{1 \pm i\sqrt{3}}{6}.$$ The value of $$81\left(x^4 + \frac{1}{6561x^4}\right) = -1,$$ not 1.