Tính biểu thức (x+ 1)/ (x-3) + (x-1)/ (x+3) - 2(x-2) (x^2) / (9-x^2)

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Tính \(\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 3}} + \frac{{{\rm{x}} - 1}}{{{\rm{x}} + 3}} - \frac{{2{\rm{x}} - 2{{\rm{x}}^2}}}{{9 - {{\rm{x}}^2}}}\).

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Giải thích

ĐKXĐ: x ≠ 3, x ¹ –3.

\(\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 3}} + \frac{{{\rm{x}} - 1}}{{{\rm{x}} + 3}} - \frac{{2{\rm{x}} - 2{{\rm{x}}^2}}}{{9 - {{\rm{x}}^2}}}\)

\( = \frac{{\left( {{\rm{x}} + 1} \right)\left( {{\rm{x}} + 3} \right)}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}} + \frac{{\left( {{\rm{x}} - 1} \right)\left( {{\rm{x}} - 3} \right)}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}} + \frac{{2{\rm{x}} - 2{{\rm{x}}^2}}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}}\)

\( = \frac{{{{\rm{x}}^2} + 4{\rm{x}} + 3 + {{\rm{x}}^2} - 4{\rm{x}} + 3 + 2{\rm{x}} - 2{{\rm{x}}^2}}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}}\)

\( = \frac{{2{\rm{x}} + 6}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}}\)\( = \frac{{2\left( {{\rm{x}} + 3} \right)}}{{\left( {{\rm{x}} + 3} \right)\left( {{\rm{x}} - 3} \right)}}\)\( = \frac{2}{{{\rm{x}} - 3}}.\)
Vậy \(\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 3}} + \frac{{{\rm{x}} - 1}}{{{\rm{x}} + 3}} - \frac{{2{\rm{x}} - 2{{\rm{x}}^2}}}{{9 - {{\rm{x}}^2}}} = \frac{2}{{{\rm{x}} - 3}}.\)