1. **Problem:** A rectangle has area 6 and perimeter 10. What is the value of its largest side? 2. **Formulas and rules:** - Area of rectangle: $A = lw$ where $l$ is length and $w$ is width. - Perimeter of rectangle: $P = 2(l + w)$. 3. **Set up equations:** Given $A = 6$ and $P = 10$, we have: $$lw = 6$$ $$2(l + w) = 10 \implies l + w = 5$$ 4. **Express one variable:** From $l + w = 5$, we get: $$w = 5 - l$$ 5. **Substitute into area equation:** $$l(5 - l) = 6$$ $$5l - l^2 = 6$$ 6. **Rewrite as quadratic:** $$-l^2 + 5l - 6 = 0$$ Multiply both sides by $-1$ to simplify: $$l^2 - 5l + 6 = 0$$ 7. **Factor quadratic:** $$l^2 - 5l + 6 = (l - 2)(l - 3) = 0$$ 8. **Solve for $l$:** $$l = 2 \quad \text{or} \quad l = 3$$ 9. **Find corresponding $w$ values:** If $l = 2$, then $w = 5 - 2 = 3$. If $l = 3$, then $w = 5 - 3 = 2$. 10. **Determine largest side:** The sides are 2 and 3, so the largest side is: $$\boxed{3}$$