\(\text{ĐKXĐ: }x-3\ge0;x+3\ge0;2x-6+\sqrt{x^2-9}\ne0\)
\(\Leftrightarrow x\ge3;x\ge-3;2x-6\ne\sqrt{x^2-9}\)
\(\Leftrightarrow x\ge3;4x^2-24x+36\ne x^2-9\)
\(\Leftrightarrow x\ge3;3x^2-24x+45\ne0\)
\(\Leftrightarrow x\ge3;3.\left(x^2-8x+15\right)\ne0\)
\(\Leftrightarrow x\ge3;\left(x-3\right)\left(x-5\right)\ne0\)
\(\Leftrightarrow x\ge3;x\ne3;x\ne5\)
\(\frac{x+3+2\sqrt{x^2-9}}{2x-6+\sqrt{x^2-9}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}.\sqrt{x+3}+2\sqrt{\left(x+3\right)\left(x-3\right)}}{2\left(x-3\right)+\sqrt{\left(x+3\right)\left(x-3\right)}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}.\sqrt{x+3}+2\sqrt{x+3}.\sqrt{x-3}}{2\sqrt{x-3}.\sqrt{x-3}+\sqrt{x+3}.\sqrt{x-3}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}\left(\sqrt{x+3}+2\sqrt{x-3}\right)}{\sqrt{x-3}\left(2\sqrt{x-3}+\sqrt{x+3}\right)}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}}{\sqrt{x-3}}=\sqrt{2}\)
\(\Leftrightarrow\frac{x+3}{x-3}=2\)
\(\Leftrightarrow x+3=2.\left(x-3\right)\)
\(\Leftrightarrow x+3=2x-6\)
\(\Leftrightarrow x-2x=-6-3\)
\(\Leftrightarrow-x=-9\)
\(\Leftrightarrow x=9\)
\(\frac{x+3+2\sqrt{x^2-9}}{2x-6+\sqrt{x^2-9}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}.\sqrt{x+3}+2\sqrt{\left(x+3\right)\left(x-3\right)}}{2\left(x-3\right)+\sqrt{\left(x+3\right)\left(x-3\right)}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}.\sqrt{x+3}+2\sqrt{x+3}.\sqrt{x-3}}{2\sqrt{x-3}.\sqrt{x-3}+\sqrt{x+3}.\sqrt{x-3}}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}\left(\sqrt{x+3}+2\sqrt{x-3}\right)}{\sqrt{x-3}\left(2\sqrt{x-3}+\sqrt{x+3}\right)}=\sqrt{2}\)
\(\Leftrightarrow\frac{\sqrt{x+3}}{\sqrt{x-3}}=\sqrt{2}\)
\(\Leftrightarrow\frac{x+3}{x-3}=2\)
\(\Leftrightarrow x+3=2.\left(x-3\right)\)
\(\Leftrightarrow x+3=2x-6\)
\(\Leftrightarrow x-2x=-6-3\)
\(\Leftrightarrow-x=-9\)
\(\Leftrightarrow x=9\)
