Thực hiện các phép tính sau:
a) \(\dfrac{{2{x^2} - 1}}{{x - 2}} + \dfrac{{ - {x^2} - 3}}{{x - 2}}\)
b) \(\dfrac{x}{{x + y}} + \dfrac{y}{{x - y}}\)
c) \(\dfrac{1}{{x - 1}} - \dfrac{2}{{{x^2} - 1}}\)
d) \(\dfrac{{x + 2}}{{{x^2} + xy}} - \dfrac{{y - 2}}{{xy + {y^2}}}\)
e) \(\dfrac{1}{{2{x^2} - 3x}} - \dfrac{1}{{4{x^2} - 9}}\)
g) \(\dfrac{{2x}}{{9 - {x^2}}} + \dfrac{1}{{x - 3}} - \dfrac{1}{{x + 3}}\)
Sử dụng quy tắc cộng, trừ phân thức
a)
\(\dfrac{{2{x^2} - 1}}{{x - 2}} + \dfrac{{ - {x^2} - 3}}{{x - 2}}\)
\( = \dfrac{{{x^2} - 4}}{{x - 2}}\)
\(\begin{array}{l} = \dfrac{{\left( {x - 2} \right)\left( {x + 2} \right)}}{{x - 2}}\\ = x + 2\end{array}\)
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b)
\(\dfrac{x}{{x + y}} + \dfrac{y}{{x - y}}\)
\(\begin{array}{l} = \dfrac{{x(x - y)}}{{(x + y)(x - y)}} + \dfrac{{y(x + y)}}{{(x - y)(x + y)}}\\ = \dfrac{{{x^2} - xy + xy + {y^2}}}{{\left( {x - y} \right)\left( {x + y} \right)}}\\ = \dfrac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}}\end{array}\)
c)
\(\dfrac{1}{{x - 1}} - \dfrac{2}{{{x^2} - 1}}\)\( = \dfrac{{x + 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} - \dfrac{2}{{\left( {x - 1} \right)\left( {x + 1} \right)}} = \dfrac{{x - 1}}{{\left( {x - 1} \right)\left( {x + 1} \right)}} = \dfrac{1}{{x + 1}}\)
d)
\(\dfrac{{x + 2}}{{{x^2} + xy}} - \dfrac{{y - 2}}{{xy + {y^2}}}\)
\(\begin{array}{l} = \dfrac{{x + 2}}{{x(x + y)}} - \dfrac{{y - 2}}{{y(x + y)}}\\ = \dfrac{{\left( {x + 2} \right)y}}{{xy\left( {x + y} \right)}} - \dfrac{{\left( {y - 2} \right)x}}{{xy\left( {x + y} \right)}}\\ = \dfrac{{xy + 2y}}{{xy\left( {x + y} \right)}} - \dfrac{{xy - 2x}}{{xy\left( {x + y} \right)}}\\ = \dfrac{{2y + 2x}}{{xy\left( {x + y} \right)}}\\ = \dfrac{{2\left( {x + y} \right)}}{{xy\left( {x + y} \right)}}\\ = \dfrac{2}{{xy}}\end{array}\)
e)
\(\dfrac{1}{{2{x^2} - 3x}} - \dfrac{1}{{4{x^2} - 9}}\)
\(\begin{array}{l} = \dfrac{1}{{x\left( {2x - 3} \right)}} - \dfrac{1}{{\left( {2x - 3} \right)\left( {2x + 3} \right)}}\\ = \dfrac{{2x + 3}}{{x\left( {2x - 3} \right)\left( {2x + 3} \right)}} - \dfrac{x}{{x\left( {2x - 3} \right)\left( {2x + 3} \right)}}\\ = \dfrac{{x + 3}}{{x\left( {4{x^2} - 9} \right)}}\end{array}\)
g)
\(\dfrac{{2x}}{{9 - {x^2}}} + \dfrac{1}{{x - 3}} - \dfrac{1}{{x + 3}}\)
\(\begin{array}{l} = \dfrac{{ - 2x}}{{\left( {x - 3} \right)\left( {x + 3} \right)}} + \dfrac{1}{{x - 3}} - \dfrac{1}{{x + 3}}\\ = \dfrac{{ - 2x}}{{\left( {x - 3} \right)\left( {x + 3} \right)}} + \dfrac{{x + 3}}{{\left( {x - 3} \right)\left( {x + 3} \right)}} - \dfrac{{x - 3}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}\\ = \dfrac{{ - 2x + 6}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}\\ = \dfrac{{ - 2\left( {x - 3} \right)}}{{\left( {x - 3} \right)\left( {x + 3} \right)}}\\ = \dfrac{{ - 2}}{{x + 3}}\end{array}\)