1. **State the problem:** Solve the equation $$9x^4 - 82x^2 + 9 = 0$$ for $x$. 2. **Use substitution:** Let $y = x^2$. Then the equation becomes a quadratic in $y$: $$9y^2 - 82y + 9 = 0$$ 3. **Apply the quadratic formula:** For $ay^2 + by + c = 0$, the solutions are $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=9$, $b=-82$, $c=9$. 4. **Calculate the discriminant:** $$\Delta = (-82)^2 - 4 \times 9 \times 9 = 6724 - 324 = 6400$$ 5. **Find the roots for $y$:** $$y = \frac{82 \pm \sqrt{6400}}{18} = \frac{82 \pm 80}{18}$$ 6. **Evaluate each root:** - For $+$ sign: $$y = \frac{82 + 80}{18} = \frac{162}{18} = 9$$ - For $-$ sign: $$y = \frac{82 - 80}{18} = \frac{2}{18} = \frac{1}{9}$$ 7. **Back-substitute $y = x^2$ and solve for $x$:** - When $x^2 = 9$, $$x = \pm 3$$ - When $x^2 = \frac{1}{9}$, $$x = \pm \frac{1}{3}$$ **Final answer:** $$x = \pm 3, \pm \frac{1}{3}$$