Xét trên miền \(x\ge1\)
\(mx+3x-2\sqrt{\left(x-1\right)\left(x+1\right)}+m-3=0\)
\(\Leftrightarrow m\left(x+1\right)=2\sqrt{\left(x-1\right)\left(x+1\right)}-3\left(x-1\right)\)
\(\Leftrightarrow m=\frac{2\sqrt{\left(x-1\right)\left(x+1\right)}-3\left(x-1\right)}{x+1}\)
\(\Leftrightarrow-3.\frac{x-1}{x+1}+2\sqrt{\frac{x-1}{x+1}}=m\)
Đặt \(\sqrt{\frac{x-1}{x+1}}=a\Rightarrow0\le a< 1\)
Xét hàm \(f\left(a\right)=-3a^2+2a\) trên \([0;1)\) ta có:
\(\left\{{}\begin{matrix}f\left(0\right)=0\\f\left(1\right)=-1\\f\left(\frac{-2}{-3.2}\right)=\frac{1}{3}\end{matrix}\right.\) \(\Rightarrow-1< f\left(a\right)\le\frac{1}{3}\) \(\forall a\in[0;1)\)
\(\Rightarrow\)Để phương trình có nghiệm \(x\ge1\) thì \(-1< m\le\frac{1}{3}\)
