a, Th1 : \(m-1=0\Rightarrow m=1\)
\(\Rightarrow-x+3=0\\ \Rightarrow x=3\)
Th2 : \(m\ne1\)
\(\Delta=\left(-1\right)^2-4.\left(m-1\right).3\\ =1-12m+12\\=13-12m \)
phương trình có nghiệm \(\Delta\ge0\)
\(\Rightarrow13-12m\ge0\\ \Rightarrow m\le\dfrac{13}{12}\)
b, Áp dụng hệ thức vi ét : \(\left\{{}\begin{matrix}x_1+x_2=\dfrac{1}{m-1}\\x_1x_1=\dfrac{3}{m-1}\end{matrix}\right.\)
Tổng bình phương hai nghiệm bằng 12 \(\Rightarrow x^2_1+x^2_2=12\)
\(\left(x_1+x_2\right)^2-2x_1x_2=12\\ \Leftrightarrow\left(\dfrac{1}{m-1}\right)^2-2.\left(\dfrac{3}{m-1}\right)=12\\ \Leftrightarrow\dfrac{1}{\left(m-1\right)^2}-\dfrac{6}{m-1}=12\\ \Leftrightarrow1-6\left(m-1\right)=12\left(m-1\right)^2\\ \Leftrightarrow1-6m+6=12\left(m^2-2m+1\right)\\ \Leftrightarrow7-6m-12m^2+24m-12=0\\ \Leftrightarrow-12m^2+18m-5=0\\ \Leftrightarrow\left[{}\begin{matrix}m=\dfrac{9-\sqrt{21}}{12}\\m=\dfrac{9+\sqrt{21}}{12}\end{matrix}\right.\Rightarrow m=\dfrac{9+\sqrt{21}}{12}\)
