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Home » Why Regress Is the Opposite of Improve — And Why It Matters - Tacotoon

Why Regress Is the Opposite of Improve — And Why It Matters - Tacotoon

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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#### 627,200Question: Let $ a $ and $ b $ be positive real numbers suc...
We are given:
\frac{a}{b} + \frac{b}{a} = 5
Let $ x = \frac{a}{b} $. Then $ \frac{b}{a} = \frac{1}{x} $, so:
x + \frac{1}{x} = 5
We want to compute:
\frac{a^2 + b^2}{a^2 - b^2} + \frac{a^2 - b^2}{a^2 + b^2}
Note that:
a^2 + b^2 = b^2(x^2 + 1), \quad a^2 - b^2 = b^2(x^2 - 1)
So the expression becomes:

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