Question

\(\sqrt{2x-5}-\sqrt{x+1}=x-6\)

Answer 1

ĐKXĐ: \(x\ge\frac{5}{2}\)

\(\Leftrightarrow\frac{x-6}{\sqrt{2x-5}+\sqrt{x+1}}=x-6\)

\(\Leftrightarrow\left[{}\begin{matrix}x-6=0\\\frac{1}{\sqrt{2x-5}+\sqrt{x+1}}=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=0\\\sqrt{2x-5}+\sqrt{x+1}=1\left(1\right)\end{matrix}\right.\)

Xét (1), do \(x\ge\frac{5}{2}\Rightarrow\left\{{}\begin{matrix}\sqrt{2x-5}\ge0\\\sqrt{x+1}\ge\sqrt{\frac{5}{2}+1}>1\end{matrix}\right.\)

\(\Rightarrow\sqrt{2x-5}+\sqrt{x+1}>1\Rightarrow\left(1\right)\) vô nghiệm

Vậy pt có nghiệm duy nhất \(x=6\)