Đáp án:
\(\dfrac{{24}}{9} < m < 5\)
Giải thích các bước giải:
Để phương trình có 2 nghiệm
\(\begin{array}{l}
\to \left\{ \begin{array}{l}
m - 5 \ne 0\\
{m^2} - 2m + 1 - m\left( {m - 5} \right) \ge 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m \ne 5\\
3m + 1 \ge 0
\end{array} \right.\\
\to \left\{ \begin{array}{l}
m \ge - \dfrac{1}{3}\\
m \ne 5
\end{array} \right.\\
Do:{x_1} < 2 < {x_2}\\
\to \left\{ \begin{array}{l}
{x_1} - 2 < 0\\
{x_2} - 2 > 0
\end{array} \right.\\
\to \left( {{x_1} - 2} \right)\left( {{x_2} - 2} \right) < 0\\
\to {x_1}{x_2} - 2\left( {{x_1} + {x_2}} \right) + 4 < 0\\
\to \dfrac{m}{{m - 5}} - 2\left( {\dfrac{{ - 2m + 2}}{{m - 5}}} \right) + 4 < 0\\
\to \dfrac{{m + 4m - 4 + 4m - 20}}{{m - 5}} < 0\\
\to \dfrac{{9m - 24}}{{m - 5}} < 0\\
\to \left[ \begin{array}{l}
\left\{ \begin{array}{l}
9m - 24 > 0\\
m - 5 < 0
\end{array} \right.\\
\left\{ \begin{array}{l}
9m - 24 < 0\\
m - 5 > 0
\end{array} \right.
\end{array} \right. \to \left[ \begin{array}{l}
\left\{ \begin{array}{l}
m > \dfrac{{24}}{9}\\
m < 5
\end{array} \right.\\
\left\{ \begin{array}{l}
m < \dfrac{{24}}{9}\\
m > 5
\end{array} \right.\left( l \right)
\end{array} \right.\\
\to \dfrac{{24}}{9} < m < 5\\
KL:\dfrac{{24}}{9} < m < 5
\end{array}\)
