Xét \(m=0\Rightarrow2x-1=0\Leftrightarrow x=\frac{1}{2}\left(l\right)\)
Xét \(m\ne0\)
a/ Để pt có 2 nghiệm phân biệt
\(\Leftrightarrow\Delta'>0\Leftrightarrow\left(m-1\right)^2-m\left(4m-1\right)>0\)
\(\Leftrightarrow m^2-2m+1-4m^2+m>0\)
\(\Leftrightarrow-3m^2-m+1>0\Leftrightarrow3m^2+m-1< 0\Leftrightarrow\frac{-1-\sqrt{13}}{6}< m< \frac{-1+\sqrt{13}}{6}\left(m\ne0\right)\)
\(\Rightarrow m\in\left(\frac{-1-\sqrt{13}}{6};\frac{-1+\sqrt{13}}{6}\right)\backslash\left\{0\right\}\)
b/ 2 nghiệm trái dấu \(\Leftrightarrow m\left(4m-1\right)< 0\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}m< 0\\m>\frac{1}{4}\end{matrix}\right.\\\left\{{}\begin{matrix}m>0\\m< \frac{1}{4}\end{matrix}\right.\end{matrix}\right.\Leftrightarrow0< m< \frac{1}{4}\)
c/ 2 nghiệm dương
\(\Leftrightarrow\left\{{}\begin{matrix}S>0\\P>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\frac{2\left(m-1\right)}{m}>0\\\frac{4m-1}{m}>0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}m>1\\m< 0\end{matrix}\right.\Rightarrow m\in(-\infty;0)\cup\left(1;+\infty\right)\)
d/ 2 nghiệm âm
\(\Leftrightarrow\left\{{}\begin{matrix}S< 0\\P>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\frac{2\left(m-1\right)}{m}< 0\\\frac{4m-1}{m}>0\end{matrix}\right.\Leftrightarrow...\)
