The work done by (x + 2) men in (x − 3) days equals the work done by x men in (x − 2) days (equal efficiencies). Find the value of x.

Introduction / Context:
With equal efficiencies, work is proportional to (number of men) × (number of days). Equating two such products yields a quadratic that simplifies to a linear equation once expanded and common terms cancel.

Given Data / Assumptions:

• (x + 2)(x − 3) = x(x − 2)
• Men have identical work rates.

Concept / Approach:
Expand both sides, cancel matching x^2 terms, and solve the resulting linear equation for x. Check that the resulting days are positive.

Step-by-Step Solution:
Left: (x + 2)(x − 3) = x^2 − x − 6Right: x(x − 2) = x^2 − 2xEquate: x^2 − x − 6 = x^2 − 2x ⇒ −x − 6 = −2x ⇒ x = 6

Verification / Alternative check:
Plug back: (6 + 2)(6 − 3) = 8*3 = 24 and 6*(6 − 2) = 6*4 = 24. Equality holds and days are positive.

Why Other Options Are Wrong:
4, 8, and 9 do not satisfy the equation; they create unequal work totals.

Common Pitfalls:
Arithmetic errors on expansion; ignoring feasibility (days must be positive).

Final Answer:
6