1. **State the problem:** Solve the equation $$\frac{1}{2}[2(x+1) -(x-3)^2] = \frac{1}{3}[3(x-1) -(x-3)^2].$$ 2. **Rewrite the equation:** $$\frac{1}{2}[2(x+1) -(x-3)^2] = \frac{1}{3}[3(x-1) -(x-3)^2].$$ 3. **Distribute inside the brackets:** $$\frac{1}{2}[2x + 2 - (x-3)^2] = \frac{1}{3}[3x - 3 - (x-3)^2].$$ 4. **Expand the square:** $$(x-3)^2 = x^2 - 6x + 9.$$ So, $$\frac{1}{2}[2x + 2 - (x^2 - 6x + 9)] = \frac{1}{3}[3x - 3 - (x^2 - 6x + 9)].$$ 5. **Simplify inside the brackets:** Left side: $$2x + 2 - x^2 + 6x - 9 = -x^2 + 8x - 7.$$ Right side: $$3x - 3 - x^2 + 6x - 9 = -x^2 + 9x - 12.$$ 6. **Rewrite the equation:** $$\frac{1}{2}(-x^2 + 8x - 7) = \frac{1}{3}(-x^2 + 9x - 12).$$ 7. **Multiply both sides by 6 (the least common multiple of 2 and 3) to clear denominators:** $$6 \times \frac{1}{2}(-x^2 + 8x - 7) = 6 \times \frac{1}{3}(-x^2 + 9x - 12)$$ $$3(-x^2 + 8x - 7) = 2(-x^2 + 9x - 12).$$ 8. **Distribute:** $$-3x^2 + 24x - 21 = -2x^2 + 18x - 24.$$ 9. **Bring all terms to one side:** $$-3x^2 + 24x - 21 + 2x^2 - 18x + 24 = 0$$ $$(-3x^2 + 2x^2) + (24x - 18x) + (-21 + 24) = 0$$ $$-x^2 + 6x + 3 = 0.$$ 10. **Multiply entire equation by -1 to simplify:** $$\cancel{-}1 \times (-x^2 + 6x + 3) = \cancel{-}1 \times 0$$ $$x^2 - 6x - 3 = 0.$$ 11. **Use quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$$ where $a=1$, $b=-6$, $c=-3$. 12. **Calculate discriminant:** $$b^2 - 4ac = (-6)^2 - 4(1)(-3) = 36 + 12 = 48.$$ 13. **Find roots:** $$x = \frac{6 \pm \sqrt{48}}{2} = \frac{6 \pm 4\sqrt{3}}{2} = 3 \pm 2\sqrt{3}.$$ **Final answer:** $$x = 3 + 2\sqrt{3} \quad \text{or} \quad x = 3 - 2\sqrt{3}.$$