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Find all values of x such that
\frac{x}{x - 5} = \frac{4}{x - 4} + \frac{3}{x} - 2.

 May 10, 2024
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\(\displaystyle \frac{x}{x - 5} = \frac{4}{x - 4} + \frac{3}{x} - 2\\ \dfrac x{x-5} = \dfrac{7x-12-2x(x-4)}{x(x-4)} = \dfrac{-2x^2+15x-12}{x^2 - 4x}\\ x^3 - 4x^2 = (x - 5)(-2x^2 + 15x - 12)\\ x^3 - 4x^2 = -2x^3 + 25x^2 - 87x + 60\\ 3x^3 - 29x^2 + 87x - 60 = 0 \)

 

Solving numerically (or using Cardano's formula or Tartaglia's formula) we have \(x \approx 0.97404\text{ or }x \approx 4.3463 \pm 1.2816i\)

 May 10, 2024

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