Đề bài
Thực hiện phép tính và rút gọn:
a] \[{2 \over {a + 3}} + {3 \over {[a + 3][a + 1]}}\] ;
b] \[{1 \over {{p^2} - 4p - 5}} + {{2p} \over {p - 5}}\] ;
c] \[x + {{3x} \over {x + 2}} + 2\] ;
d] \[{x \over {x + y}} + {4 \over {{x^2} + 3xy + 2{y^2}}} + {{ - 3x} \over {x + 2y}}\] .
Lời giải chi tiết
\[\eqalign{ & a]\,\,{2 \over {a + 3}} + {3 \over {\left[ {a + 3} \right]\left[ {a + 1} \right]}} \cr & \,\,\,\,\, = {{2\left[ {a + 1} \right]} \over {\left[ {a + 3} \right]\left[ {a + 1} \right]}} + {3 \over {\left[ {a + 3} \right]\left[ {a + 1} \right]}} \cr & \,\,\,\,\, = {{2a + 5} \over {\left[ {a + 3} \right]\left[ {a + 1} \right]}} \cr & b]\,\,{1 \over {{p^2} - 4p - 5}} + {{2p} \over {p - 5}} \cr & \,\,\,\,\, = {1 \over {{p^2} + p - 5p - 5}} + {{2p} \over {p - 5}} \cr & \,\,\,\, = {1 \over {p\left[ {p + 1} \right] - 5\left[ {p + 1} \right]}} + {{2p} \over {p - 5}} \cr & \,\,\,\,\, = {1 \over {\left[ {p + 1} \right]\left[ {p - 5} \right]}} + {{2p\left[ {p + 1} \right]} \over {\left[ {p - 5} \right]\left[ {p + 1} \right]}} \cr & \,\,\,\,\, = {{1 + 2{p^2} + 2p} \over {\left[ {p + 1} \right]\left[ {p - 5} \right]}} \cr & c]\,\,x + {{3x} \over {x + 2}} + 2 = {{x\left[ {x + 2} \right]} \over {x + 2}} + {{3x} \over {x + 2}} + {{2\left[ {x + 2} \right]} \over {x + 2}} \cr & \,\,\,\,\, = {{{x^2} + 2x + 3x + 2x + 4} \over {x + 2}} = {{{x^2} + 7x + 4} \over {x + 2}} \cr & d]\,\,{x \over {x + y}} + {4 \over {{x^2} + 3xy + 2{y^2}}} + {{ - 3x} \over {x + 2y}} \cr & \,\,\,\,\, = {x \over {x + y}} + {4 \over {{x^2} + xy + 2xy + 2{y^2}}} + {{ - 3x} \over {x + 2y}} \cr & \,\,\,\, = {x \over {x + y}} + {4 \over {x\left[ {x + y} \right] + 2y\left[ {x + y} \right]}} + {{ - 3x} \over {x + 2y}} \cr & \,\,\,\,\, = {x \over {x + y}} + {4 \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} + {{ - 3x} \over {x + 2y}} \cr & \,\,\,\,\, = {{x\left[ {x + 2y} \right]} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} + {4 \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} + {{ - 3x\left[ {x + y} \right]} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} \cr & \,\,\,\,\, = {{{x^2} + 2xy + 4 - 3{x^2} - 3xy} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} \cr & \,\,\,\,\, = {{ - 2{x^2} - xy + 4} \over {\left[ {x + y} \right]\left[ {x + 2y} \right]}} \cr} \]