[latex](2+ \sqrt{3} )^x+((2- \sqrt{3} )^x=4[/latex]

[latex]2- \sqrt{3} = \frac{(2- \sqrt{3})(2+ \sqrt{3}) }{2+ \sqrt{3} } =\frac{1}{2+ \sqrt{3} } [/latex] [latex](2+ \sqrt{3})^x+ \frac{1}{(2+ \sqrt{3})^x } =4 [/latex] [latex](2+ \sqrt{3} )^x=t[/latex], t>0 [latex]t+ \frac{1}{t} -4=0[/latex] t²-4t+1=0 D=2√3 t₁=2+√3 t₂=2-√3 [latex](2+ \sqrt{3} )^x=2+ \sqrt{3} [/latex] x=1 [latex](2+ \sqrt{3} )^x=2- \sqrt{3}[/latex] x=-1

[latex]2- \sqrt{3} = \frac{(2- \sqrt{3})(3+ \sqrt{3} ) )}{2+ \sqrt{3} } = \frac{1}{2+ \sqrt{3} } [/latex] [latex](2+ \sqrt{3} ) ^{x} + \frac{1}{(2+ \sqrt{3} ) ^{x}} =4[/latex] [latex](2+ \sqrt{3} )^x=b[/latex], [latex]b \ \textgreater \ 0[/latex] [latex]b^2-4b+1=0[/latex] [latex]D=2 \sqrt{3} [/latex]    D - дискриминант [latex]b _{1} =2+ \sqrt{3} [/latex] [latex]b _{2} =2- \sqrt{3} [/latex] [latex](2+ \sqrt{3} )^x=2+ \sqrt{3} [/latex] [latex]x=1[/latex] [latex](2+ \sqrt{3} )^x=2- \sqrt{3} [/latex] [latex]x=-1[/latex]