1. **State the problem:** Solve the equation $$x^5 - 10x^3 + 9x = 0$$ for $x$. 2. **Formula and rules:** To solve polynomial equations, we first try to factor the expression and then set each factor equal to zero. 3. **Factor the equation:** Notice that each term contains an $x$, so factor out $x$: $$x(x^4 - 10x^2 + 9) = 0$$ 4. **Set each factor to zero:** $$x = 0$$ and $$x^4 - 10x^2 + 9 = 0$$ 5. **Solve the quartic equation:** Let $y = x^2$, then the equation becomes: $$y^2 - 10y + 9 = 0$$ 6. **Use the quadratic formula:** $$y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} = \frac{10 \pm \sqrt{100 - 36}}{2} = \frac{10 \pm \sqrt{64}}{2}$$ $$y = \frac{10 \pm 8}{2}$$ 7. **Calculate the roots for $y$:** $$y_1 = \frac{10 + 8}{2} = \frac{18}{2} = 9$$ $$y_2 = \frac{10 - 8}{2} = \frac{2}{2} = 1$$ 8. **Back-substitute $y = x^2$:** $$x^2 = 9 \implies x = \pm 3$$ $$x^2 = 1 \implies x = \pm 1$$ 9. **List all solutions:** $$x = 0, \pm 1, \pm 3$$ **Final answer:** $$\boxed{\{ -3, -1, 0, 1, 3 \}}$$