Question

Пожалуйста......
$(2+ sqrt{3} )^{x} +(2- sqrt{3} )^{x} =14$

Answer 1

$(2+sqrt3)^{x}+(2-sqrt3)^{x}=14\\\2+sqrt3= frac{(2+sqrt3)(2-sqrt3)}{2-sqrt3} = frac{4-3}{2-sqrt3} =frac{1}{2-sqrt3}\\\frac{1}{(2-sqrt3)^{x}}+(2-sqrt3)^{x}=14\\a=(2-sqrt3)^{x} textgreater 0; ;; ; ; frac{1}{a}+a-14=0; ;; ; frac{a^2-14a+1}{a} =0\\a^2-14a+1=0; ,; D/4=49-1=48; ,\\a_1=7-sqrt{48}=7-4sqrt3; ,; ; a_2=7+4sqrt3; ,\\(2-sqrt3)^{x}=7-4sqrt3=(2-sqrt3)^2; ; to ; ; x=2\\(2-sqrt3)^{x}=7+4sqrt3=(2+sqrt3)^2=frac{1}{(2-sqrt3)^2}=(2-sqrt3)^{-2}; to ; x=-2$

Ответ:  х=2,  х=-2 .