$\displaystyle \begin{array}{{>{\displaystyle}l}} x^{2} -( 2m-1) x+m^{2} -1=0( 1)\\ \Delta =( 2m-1)^{2} -4\left( m^{2} -1\right) =4m+5\\ a.\ ( 1) \ có\ nghiệm\ x_{1} \ x_{2} \Leftrightarrow 4m+5\geqslant 0\\ \Leftrightarrow m\geqslant -\frac{5}{4}\\ b.\ Theo\ Viet:\ \\ x_{1} +x_{2} =2m-1\ \ \ \\ \ x_{1} x_{2} =m^{2} -1\\ Ta\ co:( x_{1} -x_{2})^{2} =x_{1} -3x_{2}\\ \Leftrightarrow ( x_{1} +x_{2})^{2} -4x_{1} x_{2} =2m-1-4x_{2}\\ \Leftrightarrow ( 2m-1)^{2} -4\left( m^{2} -1\right) =2m-1-4x_{2}\\ \Leftrightarrow 4m+5=2m-1-4x_{2}\\ \Leftrightarrow x_{2} =\frac{-2m-6}{4} =-\frac{m+3}{2}\\ \Rightarrow x_{1} =2m-1+\frac{m+3}{2} =\frac{5m+1}{2}\\ x_{1} x_{2} =m^{2} -1\\ \Leftrightarrow \frac{5m+1}{2} .\frac{m+3}{2} =1-m^{2}\\ \Leftrightarrow 5m^{2} +8m+3=4-4m^{2}\\ \Leftrightarrow 9m^{2} +8m-1=0\\ \Leftrightarrow m=\frac{1}{9} \ ( tm) \ or\ m=-1\ ( tm) \end{array}$
