Đề bài
a] \[{{x + y} \over {x - y}} + {{{x^2} - 4{y^2}} \over {{x^2} - {y^2}}} - {{x - 3y} \over {x + y}}\] ;
b] \[{1 \over {2z - 3}} - {2 \over {3 - 2z}} + {{18} \over {9 - 4{z^2}}}\] ;
c] \[{1 \over {{a^2} - 5a - 6}} - {a \over {a - 6}}\] ;
d] \[{x \over {x + y}} + {4 \over {{x^2} + 3xy + 2{y^2}}} - {{3x} \over {x + 2y}}\] .
Lời giải chi tiết
\[\eqalign{ & a]\,\,{{x + y} \over {x - y}} + {{{x^2} - 4{y^2}} \over {{x^2} - {y^2}}} - {{x - 3y} \over {x + y}} \cr & = {{x + y} \over {x - y}} + {{{x^2} - 4{y^2}} \over {\left[ {x - y} \right]\left[ {x + y} \right]}} - {{x - 3y} \over {x + y}} \cr & = {{{{\left[ {x + y} \right]}^2}} \over {\left[ {x - y} \right]\left[ {x + y} \right]}} + {{{x^2} - 4{y^2}} \over {\left[ {x - y} \right]\left[ {x + y} \right]}} - {{\left[ {x - 3y} \right]\left[ {x - y} \right]} \over {\left[ {x - y} \right]\left[ {x + y} \right]}} \cr & = {{{{\left[ {x + y} \right]}^2} + {x^2} - 4{y^2} - \left[ {x - 3y} \right]\left[ {x - y} \right]} \over {\left[ {x - y} \right]\left[ {x + y} \right]}} \cr & = {{{x^2} + 2xy + {y^2} + {x^2} - 4{y^2} - {x^2} + xy + 3xy - 3{y^2}} \over {\left[ {x - 3y} \right]\left[ {x - y} \right]}} \cr & = {{{x^2} + 6xy - 6{y^2}} \over {\left[ {x - 3y} \right]\left[ {x - y} \right]}} \cr & b]\,\,{1 \over {2z - 3}} - {2 \over {3 - 2z}} + {{18} \over {9 - 4{z^2}}} \cr & = {1 \over {2z - 3}} - {2 \over {3 - 2z}} + {{18} \over {\left[ {3 - 2z} \right]\left[ {3 + 2z} \right]}} \cr & = {{ - 3} \over {3 - 2z}} + {{18} \over {\left[ {3 - 2z} \right]\left[ {3 + 2z} \right]}} = {{ - 3\left[ {3 + 2z} \right] + 18} \over {\left[ {3 - 2z} \right]\left[ {3 + 2z} \right]}} \cr & = {{9 - 6z} \over {\left[ {3 - 2z} \right]\left[ {3 + 2z} \right]}} = {{3\left[ {3 - 2z} \right]} \over {\left[ {3 - 2z} \right]\left[ {3 + 2z} \right]}} = {3 \over {3 + 2z}} \cr & c]\,\,{1 \over {{a^2} - 5a - 6}} - {a \over {a - 6}} \cr & = {1 \over {\left[ {a + 1} \right]\left[ {a - 6} \right]}} - {a \over {a - 6}} \cr & = {{1 - a\left[ {a + 1} \right]} \over {\left[ {a + 1} \right]\left[ {a - 6} \right]}} = {{ - {a^2} - a + 1} \over {\left[ {a + 1} \right]\left[ {a - 6} \right]}} \cr & d]\,\,{x \over {x + y}} + {4 \over {{x^2} + 3xy + 2{y^2}}} + {{ - 3x} \over {x + 2y}} \cr & = {x \over {x + y}} + {4 \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} + {{ - 3x} \over {x + 2y}} \cr & = {{x\left[ {x + 2y} \right] + 4 - 3x\left[ {x + y} \right]} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} \cr & = {{{x^2} + 2xy + 4 - 3{x^2} - 3xy} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} = {{ - 2{x^2} - xy + 4} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} \cr} \]