Thực hiện phép tính (1 / (x^2 + x) - (2 - x) / (x + 1)) : (1/x + x - 2)
Thực hiện phép tính:
\(\left( {\frac{1}{{{x^2} + x}} - \frac{{2 - x}}{{x + 1}}} \right):\left( {\frac{1}{x} + x - 2} \right)\);
\(\left( {\frac{1}{{{x^2} + x}} - \frac{{2 - x}}{{x + 1}}} \right):\left( {\frac{1}{x} + x - 2} \right)\)
\( = \left[ {\frac{1}{{x\left( {x + 1} \right)}} - \frac{{2 - x}}{{x + 1}}} \right]:\left( {\frac{1}{x} + \frac{{{x^2}}}{x} - \frac{{2x}}{x}} \right)\)
\( = \left[ {\frac{1}{{x\left( {x + 1} \right)}} - \frac{{x\left( {2 - x} \right)}}{{x\left( {x + 1} \right)}}} \right]:\left( {\frac{{1 + {x^2} - 2x}}{x}} \right)\)
\( = \left[ {\frac{{1 - x\left( {2 - x} \right)}}{{x\left( {x + 1} \right)}}} \right]:\left[ {\frac{{{{\left( {x - 1} \right)}^2}}}{x}} \right]\)
\( = \frac{{1 - 2x + {x^2}}}{{x\left( {x + 1} \right)}}.\frac{x}{{{{\left( {x - 1} \right)}^2}}}\)
\( = \frac{{{{\left( {x - 1} \right)}^2}.x}}{{x.{{\left( {x - 1} \right)}^2}\left( {x + 1} \right)}}\)
\( = \frac{1}{{x + 1}}\).