Đề bài
Tìm mẫu thức chung rồi thực hiện phép tính:
a] \[{4 \over {2 + y}} + {2 \over {y - 2}} + {{5y - 6} \over {4 - {y^2}}}\] ;
b] \[{{1 - 3m} \over {2m}} + {{3m - 2} \over {2m - 1}} + {{3m - 2} \over {2m - 4{m^2}}}\] ;
c] \[{k \over {{k^2} - 9}} + {1 \over {6k - 9 - {k^2}}} + {1 \over {{k^2} + 6k + 9}}\] .
Lời giải chi tiết
\[\eqalign{ & a]\,\,{4 \over {2 + y}} + {2 \over {y - 2}} + {{5y - 6} \over {4 - {y^2}}} \cr & \,\,\,\,\, = {4 \over {2 + y}} + {{ - 2} \over {2 - y}} + {{5y - 6} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} \cr & \,\,\,\,\, = {{4\left[ {2 - y} \right]} \over {\left[ {2 + y} \right]\left[ {2 - y} \right]}} + {{ - 2\left[ {2 + y} \right]} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} + {{5y - 6} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} \cr & \,\,\,\,\, = {{8 - 4y - 4 - 2y + 5y - 6} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} \cr & \,\,\,\,\, = {{ - y - 2} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} = {{ - \left[ {y + 2} \right]} \over {\left[ {2 - y} \right]\left[ {2 + y} \right]}} = {{ - 1} \over {2 - y}} \cr & b]\,\,{{1 - 3m} \over {2m}} + {{3m - 2} \over {2m - 1}} + {{3m - 2} \over {2m - 4{m^2}}} \cr & \,\,\,\,\, = {{1 - 3m} \over {2m}} + {{ - \left[ {3m - 2} \right]} \over {1 - 2m}} + {{3m - 2} \over {2m\left[ {1 - 2m} \right]}} \cr & \,\,\,\,\, = {{\left[ {1 - 3m} \right]\left[ {1 - 2m} \right]} \over {2m\left[ {1 - 2m} \right]}} + {{ - \left[ {3m - 2} \right].2m} \over {2m\left[ {1 - 2m} \right]}} + {{3m - 2} \over {2m\left[ {1 - 2m} \right]}} \cr & \,\,\,\,\, = {{1 - 2m - 3m + 6{m^2} - 6{m^2} + 4m + 3m - 2} \over {2m\left[ {1 - 2m} \right]}} \cr & \,\,\,\,\, = {{2m - 1} \over {2m\left[ {1 - 2m} \right]}} = {{ - \left[ {1 - 2m} \right]} \over {2m\left[ {1 - 2m} \right]}} = {{ - 1} \over {2m}} \cr & c]\,\,{k \over {{k^2} - 9}} + {1 \over {6k - 9 - {k^2}}} + {1 \over {{k^2} + 6k + 9}} \cr & \,\,\,\,\, = {k \over {\left[ {k - 3} \right]\left[ {k + 3} \right]}} + {{ - 1} \over {{{\left[ {k - 3} \right]}^2}}} + {1 \over {{{\left[ {k + 3} \right]}^2}}} \cr & \,\,\,\,\, = {{k\left[ {k - 3} \right]\left[ {k + 3} \right]} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} + {{ - {{\left[ {k + 3} \right]}^2}} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} + {{{{\left[ {k - 3} \right]}^2}} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} \cr & \,\,\,\,\, = {{k\left[ {{k^2} - 9} \right] - \left[ {{k^2} + 6k + 9} \right] + \left[ {{k^2} - 6k + 9} \right]} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} \cr & \,\,\,\,\, = {{{k^3} - 9k - {k^2} - 6k - 9 + {k^2} - 6k + 9} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} \cr & \,\,\,\,\, = {{{k^3} - 21k} \over {{{\left[ {k - 3} \right]}^2}{{\left[ {k + 3} \right]}^2}}} \cr} \]