Đề bài

Thực hiện các phép tính sau: a) \(\frac{1}{{2x - 3}} - \frac{2}{{3 - 2x}} + \frac{{18}}{{9 - 4{x^2}}}\) b) \(\frac{2}{{{a^2} - 1}} - \frac{1}{{a + 1}} - \frac{1}{{a - 1}}\) c) \(\frac{{a + b}}{{a - b}} + \frac{{{a^2} - 4{b^2}}}{{{a^2} - {b^2}}} - \frac{{a - 3b}}{{a + b}}\) d) \(\frac{x}{{{x^2} + xy}} - \frac{y}{{{x^2} - {y^2}}} + \frac{{x + y}}{{xy - {y^2}}}\)

Lời giải chi tiết

a)\(\begin{array}{l}\frac{1}{{2x - 3}} - \frac{2}{{3 - 2x}} + \frac{{18}}{{9 - 4{x^2}}} = \frac{{ - \left( {3 + 2x} \right) - 2\left( {3 + 2x} \right) + 18}}{{9 - 4{x^2}}} = \frac{{ - 3 - 2x - 6 - 4x + 18}}{{9 - 4{x^2}}}\\ = \frac{{9 - 6x}}{{9 - 4{x^2}}} = \frac{{3\left( {3 - 2x} \right)}}{{\left( {3 - 2x} \right)\left( {3 + 2x} \right)}} = \frac{3}{{3 + 2x}}\end{array}\)b)\(\frac{2}{{{a^2} - 1}} - \frac{1}{{a + 1}} - \frac{1}{{a - 1}} = \frac{{2 - \left( {a - 1} \right) - \left( {a + 1} \right)}}{{\left( {a - 1} \right)\left( {a + 1} \right)}} = \frac{{2 - a + 1 - a - 1}}{{\left( {a - 1} \right)\left( {a + 1} \right)}} = \frac{{2\left( {1 - a} \right)}}{{\left( {a - 1} \right)\left( {a + 1} \right)}} = \frac{{ - 2}}{{a + 1}}\)c) \(\begin{array}{l}\frac{{a + b}}{{a - b}} + \frac{{{a^2} - 4{b^2}}}{{{a^2} - {b^2}}} - \frac{{a - 3b}}{{a + b}} = \frac{{{a^2} + 2ab + {b^2}}}{{{a^2} - {b^2}}} + \frac{{{a^2} - 4{b^2}}}{{{a^2} - {b^2}}} - \frac{{{a^2} - 4ab + 3{b^2}}}{{{a^2} - {b^2}}}\\ = \frac{{{a^2} + 6ab - 6{b^2}}}{{{a^2} - {b^2}}}\end{array}\) d)\(\begin{array}{l}\frac{x}{{{x^2} + xy}} - \frac{y}{{{x^2} - {y^2}}} + \frac{{x + y}}{{xy - {y^2}}} = \frac{x}{{x\left( {x + y} \right)}} - \frac{y}{{\left( {x + y} \right)\left( {x - y} \right)}} + \frac{{x + y}}{{y\left( {x - y} \right)}}\\ = \frac{{xy\left( {x - y} \right)}}{{xy\left( {{x^2} - {y^2}} \right)}} - \frac{{x{y^2}}}{{xy\left( {{x^2} - {y^2}} \right)}} + \frac{{x{{\left( {x + y} \right)}^2}}}{{xy\left( {{x^2} - {y^2}} \right)}}\\ = \frac{{{x^2}y - x{y^2} - x{y^2} + {x^3} + 2{x^2}y + x{y^2}}}{{xy\left( {{x^2} - {y^2}} \right)}}\\ = \frac{{{x^3} + {x^2}y - x{y^2}}}{{xy\left( {{x^2} - {y^2}} \right)}} = \frac{{{x^2} + xy - {y^2}}}{{y\left( {{x^2} - {y^2}} \right)}}\end{array}\)