ĐKXĐ: ...
Đặt \(\left\{{}\begin{matrix}\sqrt{x-1}=a\ge0\\2x-5=b\end{matrix}\right.\) \(\Rightarrow4x^2-15x+20=b^2+5a^2\)
Phương trình trở thành:
\(\sqrt{b^2+5a^2}=2b+7a\) (\(2b+7a\ge0\))
\(\Leftrightarrow b^2+5a^2=\left(2b+7a\right)^2\)
\(\Leftrightarrow44a^2+28ab+3b^2=0\)
\(\Leftrightarrow\left(22a+3b\right)\left(2a+b\right)=0\)
- Nếu \(22a+3b=0\Rightarrow b=-\frac{22}{3}a\Rightarrow2a+7b=2a-7.\frac{22}{3}a< 0\left(l\right)\)
- Nếu \(2a+b=0\Rightarrow b=-2a\Rightarrow2b+7a=5a>0\) thỏa mãn
Khi đó ta có:
\(2a=-b\Leftrightarrow2\sqrt{x-1}=5-2x\) (\(x\le\frac{5}{2}\))
\(\Leftrightarrow4\left(x-1\right)=\left(5-2x\right)^2\)
\(\Leftrightarrow4x^2-24x+29=0\Rightarrow\left[{}\begin{matrix}x=\frac{6+\sqrt{7}}{2}\left(l\right)\\x=\frac{6-\sqrt{7}}{2}\end{matrix}\right.\)