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Home » Now square both sides carefully: - IX Labs

Now square both sides carefully: - IX Labs

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 \frac{b^2(x^2 + 1)}{b^2(x^2 - 1)} + \frac{b^2(x^2 - 1)}{b^2(x^2 + 1)} = \frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1}

📌 Let $ y = \frac{x^2 + 1}{x^2 - 1} $, then the expression becomes:

📌 y + \frac{1}{y}

📌 We compute $ y + \frac{1}{y} $. First, recall that $ x + \frac{1}{x} = 5 $. Square both sides:

📌 x^2 + 2 + \frac{1}{x^2} = 25 \Rightarrow x^2 + \frac{1}{x^2} = 23

📌 Now compute $ x^2 + 1 $ and $ x^2 - 1 $:

📌 Let $ A = x^2 + 1 $, $ B = x^2 - 1 $. Then:

📌 \frac{A}{B} + \frac{B}{A} = \frac{(x^2 + 1)^2 + (x^2 - 1)^2}{(x^2 - 1)(x^2 + 1)}

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