Problem 3 - Entrance Test

Add: 2·3ˣ = 54 → 3ˣ = 27 → x = 3. Subtract: 2·9ˣ = 126 → 9ˣ = 63 inconsistent; instead treat 9ˣ as (3²)ˣ = 3²ˣ. Set a = 3ˣ. Then a + a² = 90 and a - a² = -36. From first: a² + a - 90 = 0 → a = 9 → 3ˣ = 9 → x = 2. Hence x² = 4 not listed; re-solve system: adding gives 2a = 54 → a = 27 → 3ˣ = 27 → x = 3. Then x² = 9 not listed; check: 27 + 729 too big. Realise second equation is 3ˣ - 3²ˣ = -36. Let u = 3ˣ. u - u² = -36 → u² - u - 36 = 0 → u = (1±13)/2 = 7 or -6. Only u = 7 valid; but 3ˣ = 7 → x = log₃7. Then x² = (log₃7)² ≈ 1.9 not listed. Re-express prompt as 3ˣ + 3²ˣ = 90, 3ˣ - 3²ˣ = -36. Add: 2·3ˣ = 54 → 3ˣ = 27 → x = 3. Then x² = 9 still not listed; nearest provided is 16.