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Home » Experience the Magic of Sunset Orange—Nature’s Ultimate Masterpiece! - DNSFLEX

Experience the Magic of Sunset Orange—Nature’s Ultimate Masterpiece! - DNSFLEX

📅 March 12, 2026 👤 scraface

Mar 12, 2026

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📌 \frac{2(x^2 + 1)}{x^2 - 1}

📌 From earlier, $ x + \frac{1}{x} = 5 \Rightarrow x^2 + \frac{1}{x^2} = 23 $, so:

📌 x^2 + 1 = x^2 + \frac{x^2}{x^2} = x^2 + 1 \Rightarrow \text{Try direct substitution}

📌 Let $ x = \frac{a}{b} $, then $ x + \frac{1}{x} = 5 \Rightarrow x^2 + \frac{1}{x^2} = 23 $, so:

📌 x^4 + 1 = x^4 + \frac{1}{x^4} \cdot x^4 = x^4 + 1 \Rightarrow \text{Not helpful}

📌 Instead, use:

📌 \frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1} = \frac{(x^2 + 1)^2 + (x^2 - 1)^2}{(x^2 - 1)(x^2 + 1)} = \frac{2x^4 + 2}{x^4 - 1} = 2 \cdot \frac{x^4 + 1}{x^4 - 1}

📌 Let’s compute $ x^4 + 1 $, $ x^4 - 1 $:

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