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Home » For (1,4): \( a + b + c = 4 \) - Midis

For (1,4): \( a + b + c = 4 \) - Midis

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 f(f(x)) = rac{(N/D)^3 - 3(N/D)}{(N/D)^2 + 1} = rac{ rac{N^3 - 3ND^2}{D^3} }{ rac{N^2 + D^2}{D^2} } = rac{N^3 - 3ND^2}{D^3} \cdot rac{D^2}{N^2 + D^2} = rac{N^3 - 3ND^2}{D(N^2 + D^2)}

📌 So, \( f(f(x)) = x \) becomes:

📌 rac{N^3 - 3ND^2}{D(N^2 + D^2)} = x

📌 Multiply both sides by \( D(N^2 + D^2) \):

📌 N^3 - 3ND^2 = x D (N^2 + D^2)

📌 Now substitute back \( N = x^3 - 3x \), \( D = x^2 + 1 \):

📌 \( N^2 = (x^3 - 3x)^2 = x^6 - 6x^4 + 9x^2 \)

📌 \( N^3 = (x^3 - 3x)^3 = x^9 - 9x^7 + 27x^5 - 27x^3 \)

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x^4 + 1 = x^4 + \frac{1}{x^4} \cdot x^4 = x^4 + \frac{x^4}{x^4} = x^4...
Instead, observe:
\frac{x^2 + 1}{x^2 - 1} + \frac{x^2 - 1}{x^2 + 1} = y + \frac{1}{y} =...
Let’s compute this directly using the identity:
\frac{a + b}{a - b} + \frac{a - b}{a + b} = \frac{(a + b)^2 + (a - b)^...
\frac{2(x^2 + 1)}{x^2 - 1}
From earlier, $ x + \frac{1}{x} = 5 \Rightarrow x^2 + \frac{1}{x^2} =...
x^2 + 1 = x^2 + \frac{x^2}{x^2} = x^2 + 1 \Rightarrow \text{Try direct...
Let $ x = \frac{a}{b} $, then $ x + \frac{1}{x} = 5 \Rightarrow x^2 +...
x^4 + 1 = x^4 + \frac{1}{x^4} \cdot x^4 = x^4 + 1 \Rightarrow \text{No...

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