ĐKXĐ : \(x\ge\sqrt{3}\)
\(\sqrt{3x+\sqrt{3}}-\sqrt{x-\sqrt{3}}=2\sqrt{x}\)
\(\Leftrightarrow3x+\sqrt{3}-2\sqrt{\left(3x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}+x-\sqrt{3}=4x\)
\(\Leftrightarrow2\sqrt{\left(3x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}=0\)
\(\Leftrightarrow\orbr{\begin{cases}3x+\sqrt{3}=0\\x-\sqrt{3}=0\end{cases}\Leftrightarrow\orbr{\begin{cases}x=\frac{-\sqrt{3}}{3}\left(ktm\right)\\x=\sqrt{3}\left(tm\right)\end{cases}}}\)
Vậy phương trình có nghiệm duy nhất là \(x=\sqrt{3}\)
đk: \(x\ge\sqrt{3}\)
Ta có: \(\sqrt{3x+\sqrt{3}}-\sqrt{x-\sqrt{3}}=2\sqrt{x}\)
\(\Leftrightarrow3x+\sqrt{3}-2\sqrt{\left(3x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}+x-\sqrt{3}=4x\)
\(\Leftrightarrow2\sqrt{\left(3x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}=0\)
\(\Leftrightarrow\left(3x+\sqrt{3}\right)\left(x-\sqrt{3}\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}3x+\sqrt{3}=0\\x-\sqrt{3}=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=-\frac{\sqrt{3}}{3}\left(ktm\right)\\x=\sqrt{3}\left(tm\right)\end{cases}}\)
Vậy \(x=\sqrt{3}\)
ĐKXĐ: \(x\ge\sqrt{3}\)
\(\sqrt{3x+\sqrt{3}}=2\sqrt{x}+\sqrt{x-\sqrt{3}}\)
+) Xét \(2\sqrt{x}=\sqrt{x-\sqrt{3}}\Rightarrow4x=x-3\Leftrightarrow x=-1\)---> Không thỏa ĐKXĐ
Vậy \(2\sqrt{x}-\sqrt{x-\sqrt{3}}\ne0\)---> Ta dùng lượng liên hiệp:
\(\sqrt{3x+\sqrt{3}}=\frac{\left(2\sqrt{x}+\sqrt{x-\sqrt{3}}\right)\left(2\sqrt{x}-\sqrt{x-\sqrt{3}}\right)}{2\sqrt{x}-\sqrt{x-\sqrt{3}}}=\frac{4x-\left(x-\sqrt{3}\right)}{2\sqrt{x}-\sqrt{x-\sqrt{3}}}\)
\(\sqrt{3x+\sqrt{3}}=\frac{3x+\sqrt{3}}{2\sqrt{x}-\sqrt{x-\sqrt{3}}}\Leftrightarrow\sqrt{3x+\sqrt{3}}\left(1-\frac{\sqrt{3x+\sqrt{3}}}{2\sqrt{x}-\sqrt{x-\sqrt{3}}}\right)=0\)
Vì \(x\ge\sqrt{3}\Rightarrow\sqrt{3x+\sqrt{3}}>0\Rightarrow1-\frac{\sqrt{3x+\sqrt{3}}}{2\sqrt{x}-\sqrt{x-\sqrt{3}}}=0\)
\(\Leftrightarrow2\sqrt{x}-\sqrt{x-\sqrt{3}}=\sqrt{3x+\sqrt{3}}\Rightarrow3x+\sqrt{3}-4\sqrt{x}.\sqrt{x-\sqrt{3}}=3x+\sqrt{3}\)
\(\Leftrightarrow\sqrt{x}.\sqrt{x-\sqrt{3}}=0\Rightarrow\orbr{\begin{cases}x=0\\x=\sqrt{3}\end{cases}}\)
Vì x = 0 không thỏa ĐKXĐ vậy PT nhận nghiệm duy nhất là \(x=\sqrt{3}\)