1. **State the problem:** Solve the equation $$\frac{x}{3} \cdot \frac{x - 2}{2} = \frac{13}{5}$$. 2. **Write the formula and rules:** We need to solve for $x$ by first simplifying the left side and then isolating $x$. Multiplying fractions means multiplying numerators and denominators. 3. **Multiply the fractions on the left:** $$\frac{x}{3} \cdot \frac{x - 2}{2} = \frac{x(x - 2)}{3 \cdot 2} = \frac{x(x - 2)}{6}$$ 4. **Rewrite the equation:** $$\frac{x(x - 2)}{6} = \frac{13}{5}$$ 5. **Cross multiply to clear denominators:** $$5 \cdot x(x - 2) = 6 \cdot 13$$ 6. **Simplify both sides:** $$5x(x - 2) = 78$$ 7. **Expand the left side:** $$5(x^2 - 2x) = 78$$ $$5x^2 - 10x = 78$$ 8. **Bring all terms to one side to form a quadratic equation:** $$5x^2 - 10x - 78 = 0$$ 9. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=5$, $b=-10$, $c=-78$. 10. **Calculate the discriminant:** $$b^2 - 4ac = (-10)^2 - 4 \cdot 5 \cdot (-78) = 100 + 1560 = 1660$$ 11. **Calculate the roots:** $$x = \frac{10 \pm \sqrt{1660}}{10}$$ 12. **Simplify the square root:** $$\sqrt{1660} = \sqrt{4 \cdot 415} = 2\sqrt{415}$$ 13. **Final solutions:** $$x = \frac{10 \pm 2\sqrt{415}}{10} = 1 \pm \frac{\sqrt{415}}{5}$$ **Answer:** $$x = 1 + \frac{\sqrt{415}}{5} \quad \text{or} \quad x = 1 - \frac{\sqrt{415}}{5}$$