Shortcut Trick To get the terms 2x and 1/(3x), multiply the given equation by 2/3. ⇒ (2/3) × [3x − 1/(2x)] = 4 × (2/3) ⇒ 2x − 1/(3x) = 8/3 Direct formula: If ax − 1/(bx) = k, then a^(2)x^(2) + 1/(b^(2)x^(2)) = k^(2) + 2a/b ⇒ 4x^(2) + 1/(9x^(2)) = (8/3)^(2) + 2 × (2/3) = 64/9 + 4/3 = 64/9 + 12/9 ⇒ 4x^(2) + 1/(9x^(2)) = 76/9 ∴ The correct answer is 76/9. Alternate Method Given: 3x − 1/(2x) = 4 Formula Used: (a − b)^(2) = a^(2) + b^(2) − 2ab Calculations: We need to find the value of 4x^(2) + 1/(9x^(2)), which can be written as (2x)^(2) + 1/(3x)^(2). Multiply the given equation by 2/3 on both sides to match the coefficients: ⇒ (2/3) × [3x − 1/(2x)] = 4 × (2/3) ⇒ 2x − 1/(3x) = 8/3 Squaring both sides, we get: ⇒ [2x − 1/(3x)]^(2) = (8/3)^(2) ⇒ (2x)^(2) + 1/(3x)^(2) − 2 × (2x) × [1/(3x)] = 64/9 ⇒ 4x^(2) + 1/(9x^(2)) − 4/3 = 64/9 ⇒ 4x^(2) + 1/(9x^(2)) = 64/9 + 4/3 ⇒ 4x^(2) + 1/(9x^(2)) = (64 + 12) / 9 ⇒ 4x^(2) + 1/(9x^(2)) = 76/9 ∴ The correct answer is 76/9. Related Concepts: Algebraic Identities (a + b)^(2) = a^(2) + 2ab + b^(2) Important Deductions If x + 1/x = k, then x^(2) + 1/x^(2) = k^(2) − 2 Inverse Term Squares If ax − 1/(bx) = k, then a^(2)x^(2) + 1/(b^(2)x^(2)) = k^(2) + 2a/b