Question

Giải phương trình:

\(\sqrt{1-x}+\sqrt{x^2-3x+2}+\left(x-2\right)\sqrt{\frac{x-1}{x-2}}=3\)

Answer 1

Đk: \(\hept{\begin{cases}1-x\ge0\\x^2-3x+2\ge0\\\frac{x-1}{x-2}\ge0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\le1\\x\le1\vee x\ge2\\x\le1\vee x\ge2\end{cases}\Leftrightarrow x\le1.}\)

Khi đó \(x-1\le0\Rightarrow x-2\le-1< 0\)

Đặt \(\hept{\begin{cases}\sqrt{1-x}=a\\\sqrt{2-x}=b\end{cases}\Rightarrow\hept{\begin{cases}b^2-a^2=1\\a+ab+\frac{\left(-b^2\right)a}{b}=3\left(1\right)\end{cases}}}\)

Từ (1) \(a+ab-ab=3\Rightarrow a=3\)

\(\Rightarrow\sqrt{1-x}=3\Rightarrow1-x=9\Rightarrow x=-8.\)