Question

Giải phương trình

\(7+2\sqrt{x}-x=\left(2+\sqrt{x}\right)\sqrt{7-x}\)

Answer 1

\(0\le x\le7\)

\(7-x+2\sqrt{x}-2\sqrt{7-x}-\sqrt{x}\sqrt{7-x}=0\)

\(\Leftrightarrow\sqrt{7-x}\left(\sqrt{7-x}-2\right)-\sqrt{x}\left(\sqrt{7-x}-2\right)=0\)

\(\Leftrightarrow\left(\sqrt{7-x}-\sqrt{x}\right)\left(\sqrt{7-x}-2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{7-x}=\sqrt{x}\\\sqrt{7-x}=2\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}7-x=x\\7-x=4\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=\frac{7}{2}\\x=3\end{matrix}\right.\)