[tex]\frac{2-\sqrt{2} }{3\sqrt{3}-x } + 2\sqrt{2} = 4\\\\x \neq 3\sqrt{3} \\\\\frac{2-\sqrt{2} }{3\sqrt{3}-x } = 4-2\sqrt{2}\\2-\sqrt{2} = 2(2-\sqrt{2})(3\sqrt{3}-x)\\2 - \sqrt{2} = 2(6\sqrt{3} -2x -3\sqrt{6}+x\sqrt{2})\\2-\sqrt{2} = 12\sqrt{3} - 4x - 6\sqrt{6}+ 2x\sqrt{2}-2\\4x - 2x\sqrt{2} = 12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2\\x(4-2\sqrt{2}) = 12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2\\[/tex]
[tex]x = \frac{12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2}{4-2\sqrt{2}} = \frac{(12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2)(4+2\sqrt{2}) }{(4-2\sqrt{2})(4+ 2\sqrt{2})} = \frac{(12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2)2(2+\sqrt{2}) }{8}= \frac{(12\sqrt{3} - 6\sqrt{6} + \sqrt{2}-2)(2+\sqrt{2}) }{4}= \\\frac{24\sqrt{3}- 12\sqrt{6}+2\sqrt{2}-4+12\sqrt{6}-12\sqrt{3}+2-2\sqrt{2}}{4}= \frac{12\sqrt{3}+2}{4} = \frac{6\sqrt{3}+1}{2}[/tex]
