Đáp án: $m = \dfrac{{89}}{{36}}$
Giải thích các bước giải:
$\begin{array}{l}
{x^2} - 2mx + {m^2} - 2 = 0\\
\Delta ' \ge 0\\
\Rightarrow {m^2} - {m^2} + 2 \ge 0\\
\Rightarrow 2 \ge 0\left( {tm} \right)\\
Theo\,Viet:\left\{ \begin{array}{l}
{x_1} + {x_2} = 2m\\
{x_1}{x_2} = {m^2} - 2
\end{array} \right.\\
Dk:{x_1};{x_2} \ge 0\\
\Rightarrow \left\{ \begin{array}{l}
2m \ge 0\\
{m^2} - 2 \ge 0
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
m \ge 0\\
{m^2} \ge 2
\end{array} \right.\\
\Rightarrow m \ge \sqrt 2 \\
Do:\sqrt {{x_1}} + \sqrt {{x_2}} = 3\\
\Rightarrow {x_1} + 2\sqrt {{x_1}{x_2}} + {x_2} = 9\\
\Rightarrow 2m + 2\sqrt {{m^2} - 2} = 9\\
\Rightarrow 2\sqrt {{m^2} - 2} = 9 - 2m\left( {dk:m \le \dfrac{9}{2}} \right)\\
\Rightarrow 4\left( {{m^2} - 2} \right) = {\left( {9 - 2m} \right)^2}\\
\Rightarrow 4{m^2} - 8 = 4{m^2} - 36m + 81\\
\Rightarrow 36m = 89\\
\Rightarrow m = \dfrac{{89}}{{36}}\left( {tmdk} \right)
\end{array}$
