Đáp án:
\(\left[ \begin{array}{l}
m = 0\\
m = \dfrac{{32}}{5}
\end{array} \right.\)
Giải thích các bước giải:
\(\begin{array}{l}
DK:9{m^2} + 24m + 16 - 4\left( {2{m^2} + 5m + 3} \right) \ge 0\\
\to 9{m^2} + 24m + 16 - 8{m^2} - 10m - 12 \ge 0\\
\to {m^2} + 14m + 4 \ge 0\\
\to {m^2} + 14m + 49 - 45 \ge 0\\
\to {\left( {m + 7} \right)^2} \ge 45\\
\to \left| {m + 7} \right| \ge 3\sqrt 5 \\
\to \left[ \begin{array}{l}
m + 7 \ge 3\sqrt 5 \\
m + 7 \le - 3\sqrt 5
\end{array} \right.\\
\to \left[ \begin{array}{l}
m \ge - 7 + 3\sqrt 5 \\
m \le - 7 - 3\sqrt 5
\end{array} \right.\\
\to \left[ \begin{array}{l}
x = \dfrac{{3m + 4 + \sqrt {{m^2} + 14m + 4} }}{2}\\
x = \dfrac{{3m + 4 - \sqrt {{m^2} + 14m + 4} }}{2}
\end{array} \right.\\
Có:{x_1} = 3{x_2}\\
\to \left[ \begin{array}{l}
\dfrac{{3m + 4 + \sqrt {{m^2} + 14m + 4} }}{2} = 3.\dfrac{{3m + 4 - \sqrt {{m^2} + 14m + 4} }}{2}\\
\dfrac{{3m + 4 - \sqrt {{m^2} + 14m + 4} }}{2} = 3.\dfrac{{3m + 4 + \sqrt {{m^2} + 14m + 4} }}{2}
\end{array} \right.\\
\to \left[ \begin{array}{l}
3m + 4 + \sqrt {{m^2} + 14m + 4} = 9m + 12 - 3\sqrt {{m^2} + 14m + 4} \\
3m + 4 - \sqrt {{m^2} + 14m + 4} = 9m + 12 + 3\sqrt {{m^2} + 14m + 4}
\end{array} \right.\\
\to \left[ \begin{array}{l}
4\sqrt {{m^2} + 14m + 4} = 6m + 8\\
4\sqrt {{m^2} + 14m + 4} = - 6m - 8
\end{array} \right.\\
\to 4\sqrt {{m^2} + 14m + 4} = 6m + 8\\
\to 2\sqrt {{m^2} + 14m + 4} = 3m + 4\\
\to 4\left( {{m^2} + 14m + 4} \right) = 9{m^2} + 24m + 16\\
\to 4{m^2} + 56m + 16 = 9{m^2} + 24m + 16\\
\to 5{m^2} - 32m = 0\\
\to \left[ \begin{array}{l}
m = 0\\
m = \dfrac{{32}}{5}
\end{array} \right.\left( {TM} \right)
\end{array}\)
