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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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xx5=4x4+3x2xx5=7x122x(x4)x(x4)=2x2+15x12x24xx34x2=(x5)(2x2+15x12)x34x2=2x3+25x287x+603x329x2+87x60=0

 

Solving numerically (or using Cardano's formula or Tartaglia's formula) we have x0.97404 or x4.3463±1.2816i

 May 10, 2024

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