[tex]\displaystyle\bf\\x^{2} +2\sqrt{41-x^{2} } =33\\\\2\sqrt{41-x^{2} } =(41-x^{2} )-8\\\\\sqrt{41-x^{2} } =m \ , \ m\geq 0 \ \ \Rightarrow \ \ 41-x^{2} =m^{2} \\\\m^{2} -2m-8=0\\\\D=(-2)^{2} -4\cdot(-8)=4+32=36=6^{2} \\\\\\m_{1} =\frac{2+6}{2} =4\\\\\\m_{2} =\frac{2-6}{2} =-2 < 0- \ ne \ podxodit\\\\\\\sqrt{41-x^{2} } =4\\\\(\sqrt{41-x^{2} } )^{2} =4^{2} \\\\41-x^{2} =16\\\\x^{2} =25\\\\x_{1,2} =\pm \ 5\\\\Proverka \ :\\\\x=5\\\\5^{2} +2\sqrt{41-25} =33\\\\25+2\sqrt{16} =33[/tex]

[tex]\displaystyle\bf\\25+2\cdot 4=33\\\\25+8=33\\\\33=33 \ - \ verno\\\\Otvet \ : \ 5[/tex]