Đáp án:
\(\left[ \begin{array}{l}
m = 2 + 2\sqrt 7 \\
m = 2 - 2\sqrt 7
\end{array} \right.\)
Giải thích các bước giải:
Để phương trình có nghiệm
\(\begin{array}{l}
\to \Delta \ge 0\\
\to {\left( {m - 2} \right)^2} - 4.\left( { - 6} \right) \ge 0\\
\to {\left( {m - 2} \right)^2} + 24 \ge 0\left( {ld} \right)\forall m\\
Vi - et:\left\{ \begin{array}{l}
{x_1} + {x_2} = m - 2\\
{x_1}{x_2} = - 6
\end{array} \right.\\
Có:{x_2}^2 - {x_1}{x_2} + \left( {m - 2} \right){x_1} = 16\\
\to {x_2}^2 - {x_1}{x_2} + \left( {{x_1} + {x_2}} \right){x_1} = 16\\
\to {x_2}^2 - {x_1}{x_2} + {x_1}^2 + {x_1}{x_2} = 16\\
\to {x_2}^2 + {x_1}^2 = 16\\
\to {x_1}^2 + 2{x_1}{x_2} + {x_2}^2 - 2{x_1}{x_2} = 16\\
\to {\left( {{x_1} + {x_2}} \right)^2} - 2{x_1}{x_2} = 16\\
\to {\left( {m - 2} \right)^2} - 2.\left( { - 6} \right) = 16\\
\to {\left( {m - 2} \right)^2} = 28\\
\to \left| {m - 2} \right| = 2\sqrt 7 \\
\to \left[ \begin{array}{l}
m - 2 = 2\sqrt 7 \\
m - 2 = - 2\sqrt 7
\end{array} \right.\\
\to \left[ \begin{array}{l}
m = 2 + 2\sqrt 7 \\
m = 2 - 2\sqrt 7
\end{array} \right.
\end{array}\)
