\(pt:x^2-4x-m^2+3=0\left(1\right)\)
\(a.\Delta=\left(-4\right)^2-3.1.\left(3-m^2\right)=16-9+3m^2=3m^2+7>0\forall m\)
⇒ pt (1) luôn có 2 nghiệm phân biệt với mọi m
Theo hệ thức Vi-et ta có:
\(\left\{{}\begin{matrix}x_1+x_2=4\left(1\right)\\x_1.x_2=3-m^2\left(2\right)\end{matrix}\right.\)
\(x_2=-5x_1\left(''\right)\)
Kết hợp :\(\left(1\right);\left(''\right)\Rightarrow\left\{{}\begin{matrix}x_1+x_2=4\\5x_1+x_2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4x_1=-4\\x_1+x_2=4\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x_1=-1\\x_2=4-x_1=4-\left(-1\right)=5\end{matrix}\right.\)
Thay \(x_1=-1;x_2=5\) vào (2)
\(\Rightarrow\left(-1\right).5=3-m^2\Leftrightarrow m^2=8\Leftrightarrow m=\pm\sqrt{8}=\pm2\sqrt{2}\)