1. **State the problem:** Solve the equation $x - 3 - 3x(x - 3) = 0$ for $x$. 2. **Rewrite the equation:** The equation is $x - 3 - 3x(x - 3) = 0$. 3. **Expand the product:** Use distributive property to expand $-3x(x - 3)$: $$-3x(x - 3) = -3x^2 + 9x$$ 4. **Substitute back:** The equation becomes: $$x - 3 - 3x^2 + 9x = 0$$ 5. **Combine like terms:** Combine $x$ and $9x$: $$-3x^2 + 10x - 3 = 0$$ 6. **Rewrite in standard quadratic form:** $$-3x^2 + 10x - 3 = 0$$ 7. **Multiply both sides by $-1$ to simplify:** $$3x^2 - 10x + 3 = 0$$ 8. **Use quadratic formula:** For $ax^2 + bx + c = 0$, solutions are $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a=3$, $b=-10$, $c=3$. 9. **Calculate discriminant:** $$\Delta = b^2 - 4ac = (-10)^2 - 4 \times 3 \times 3 = 100 - 36 = 64$$ 10. **Calculate roots:** $$x = \frac{-(-10) \pm \sqrt{64}}{2 \times 3} = \frac{10 \pm 8}{6}$$ 11. **Find each root:** - For $+$ sign: $$x = \frac{10 + 8}{6} = \frac{18}{6} = 3$$ - For $-$ sign: $$x = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$$ 12. **Final answer:** The solutions are $$x = 3 \quad \text{or} \quad x = \frac{1}{3}$$