1. **State the problem:** Solve the equation $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$ for $x$. 2. **Write the equation clearly:** $$\frac{x}{2}(x+5) - \frac{1}{3}(x-2) = 0$$ 3. **Distribute terms:** $$\frac{x}{2} \cdot (x+5) = \frac{x^2}{2} + \frac{5x}{2}$$ $$\frac{1}{3} \cdot (x-2) = \frac{x}{3} - \frac{2}{3}$$ So the equation becomes: $$\frac{x^2}{2} + \frac{5x}{2} - \left(\frac{x}{3} - \frac{2}{3}\right) = 0$$ 4. **Remove parentheses carefully:** $$\frac{x^2}{2} + \frac{5x}{2} - \frac{x}{3} + \frac{2}{3} = 0$$ 5. **Find common denominator to combine terms:** The denominators are 2 and 3, so common denominator is 6. Rewrite each term: $$\frac{x^2}{2} = \frac{3x^2}{6}$$ $$\frac{5x}{2} = \frac{15x}{6}$$ $$\frac{x}{3} = \frac{2x}{6}$$ $$\frac{2}{3} = \frac{4}{6}$$ 6. **Rewrite equation with denominator 6:** $$\frac{3x^2}{6} + \frac{15x}{6} - \frac{2x}{6} + \frac{4}{6} = 0$$ 7. **Combine like terms in numerator:** $$\frac{3x^2 + 15x - 2x + 4}{6} = 0$$ $$\frac{3x^2 + 13x + 4}{6} = 0$$ 8. **Multiply both sides by 6 to clear denominator:** $$\cancel{6} \cdot \frac{3x^2 + 13x + 4}{\cancel{6}} = 0 \cdot 6$$ $$3x^2 + 13x + 4 = 0$$ 9. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=13$, $c=4$. Calculate discriminant: $$\Delta = 13^2 - 4 \cdot 3 \cdot 4 = 169 - 48 = 121$$ 10. **Calculate roots:** $$x = \frac{-13 \pm \sqrt{121}}{2 \cdot 3} = \frac{-13 \pm 11}{6}$$ 11. **Find each solution:** - For $+$: $$x = \frac{-13 + 11}{6} = \frac{-2}{6} = -\frac{1}{3}$$ - For $-$: $$x = \frac{-13 - 11}{6} = \frac{-24}{6} = -4$$ **Final answer:** $$x = -\frac{1}{3} \quad \text{or} \quad x = -4$$