Đáp án:
a) Không có m. b) `m<22/7` c) Không có m.
Giải thích các bước giải:
\(\eqalign{ & {x^2} + \left( {2m - 6} \right)x + m - 13 = 0 \cr & \Delta = {\left( {2m - 6} \right)^2} - 4\left( {m - 13} \right) \cr & \,\,\,\, = 4{m^2} - 24m + 36 - 4m + 52 \cr & \,\,\,\, = 4{m^2} - 28m + 88 > 0\,\,\forall m \cr & \Rightarrow Pt\,\,luon\,\,co\,\,2\,\,nghiem\,\,phan\,\,biet. \cr & Ap\,\,dung\,\,dinh\,\,li\,\,Vi - et:\,\,\left\{ \matrix{ {x_1} + {x_2} = 6 - 2m \hfill \cr {x_1}{x_2} = m - 13 \hfill \cr} \right. \cr & a)\,\,{x_1}{x_2} - x_1^2 - x_2^2 = - 3 \cr & \Leftrightarrow {x_1}{x_2} - {\left( {{x_1} + {x_2}} \right)^2} + 2{x_1}{x_2} = - 3 \cr & \Leftrightarrow 3{x_1}{x_2} - {\left( {{x_1} + {x_2}} \right)^2} = - 3 \cr & \Leftrightarrow 3\left( {m - 13} \right) - {\left( {6 - 2m} \right)^2} = - 3 \cr & \Leftrightarrow 3m - 39 - 4{m^2} + 24m - 36 = - 3 \cr & \Leftrightarrow 4{m^2} - 27m + 72 = 0\,\,\left( {Vo\,\,nghiem} \right) \cr & \Rightarrow Khong\,\,co\,\,m\,\,thoa\,\,man. \cr & b)\,\,{x_1} < 3 < {x_2} \cr & \Leftrightarrow \left\{ \matrix{ {x_1} - 3 < 0 \hfill \cr {x_2} - 3 > 0 \hfill \cr} \right. \Leftrightarrow \left( {{x_1} - 3} \right)\left( {{x_2} - 3} \right) < 0 \cr & \Leftrightarrow {x_1}{x_2} - 3\left( {{x_1} + {x_2}} \right) + 9 < 0 \cr & \Leftrightarrow m - 13 - 3\left( {6 - 2m} \right) + 9 < 0 \cr & \Leftrightarrow m - 13 - 18 + 6m + 9 < 0 \cr & \Leftrightarrow 7m - 22 < 0 \Leftrightarrow m < {{22} \over 7} \cr & c)\,\,{x_1} - 3{x_2} = 5 \cr & \Leftrightarrow {x_1} = 3{x_2} + 5 \cr & \Rightarrow {x_1} + {x_2} = 4{x_2} + 5 = 6 - 2m \cr & \Leftrightarrow 4{x_2} = 1 - 2m \Leftrightarrow {x_2} = {{1 - 2m} \over 4} \cr & \Rightarrow {x_1} = {{3 - 6m} \over 4} + 5 = {{23 - 6m} \over 4} \cr & {x_1}{x_2} = m - 13 \cr & \Leftrightarrow {{23 - 6m} \over 4}{{1 - 2m} \over 4} = m - 13 \cr & \Leftrightarrow 23 - 46m - 6m + 12{m^2} = 4m - 52 \cr & \Leftrightarrow 12{m^2} - 56m + 75 = 0\,\,\left( {Vo\,\,nghiem} \right) \cr & \Rightarrow Khong\,\,co\,\,m\,\,thoa\,\,man. \cr} \)
