a.
Pt có 2 nghiệm khi \(\Delta=9-4\left(m-2\right)\ge0\Rightarrow m\le\dfrac{17}{4}\)
b.
Khi pt có 2 nghiệm, theo hệ thức Viet: \(\left\{{}\begin{matrix}x_1+x_2=3\\x_1x_2=m-2\end{matrix}\right.\)
Do \(x_1\) là nghiệm \(\Rightarrow x_1^2-3x_1+m-2=0\Rightarrow x_1^2=3x_1-m+2\)
\(\Rightarrow3x_1-m+2-x_1x_2+2x_1+x_2=m-1\)
\(\Leftrightarrow5x_1+x_2=3m-5\)
Kết hợp \(x_1+x_2=3\Rightarrow\left\{{}\begin{matrix}x_1+x_2=3\\5x_1+x_2=3m-5\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}x_1=\dfrac{3m-8}{4}\\x_2=\dfrac{-3m+20}{4}\end{matrix}\right.\)
Thay vào \(x_1x_2=m-2\)
\(\Rightarrow\left(\dfrac{3m-8}{4}\right)\left(\dfrac{-3m+20}{4}\right)=m-2\)
\(\Leftrightarrow9m^2-68m+128=0\Rightarrow\left[{}\begin{matrix}m=4\\m=\dfrac{32}{9}\end{matrix}\right.\) (thỏa mãn)
