Question

Calculați:
x/(x²-x) + (x+2)/(2+x-2x²-x³) + x²/(x²+x)

Answer 1

$\it x^2-x=x(x-1) \\ \\ 2+x-2x^2-x^3=(2+x)-x^2(2+x) =(2+x)(1-x^2) = (x+2)(1-x^2)\\ \\x^2+x=x(x+1)$

Expresia din enunț devine:

$\it E=\dfrac{x}{x-1} +\dfrac{x+2}{(x+2)(1-x^2)}+\dfrac{x^2}{x(x+1)}=\dfrac{1}{x-1}+\dfrac{1}{1-x^2}+\dfrac{x}{x+1} = \\ \\ \\ \dfrac{1}{x-1}+\dfrac{x}{x+1} +\dfrac{1}{1-x^2} = \dfrac{x+1+x^2-x}{(x-1)(x+1)} + \dfrac{1}{1-x^2} =\dfrac{x^2+1}{x^2-1}-\dfrac{1}{x^2-1}=\\ \\ \\ =\dfrac{x^2+1-1}{x^2-1}=\dfrac{x^2}{x^2-1}$

Answer 2

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