Question

$sqrt{x+sqrt{x+11}} +sqrt{x-sqrt{x+11 }} =4$
Помогите пожалуйста решить

Answer 1

$sqrt{x+sqrt{x+11}}+sqrt{x-sqrt{x+11}} =4\\ (sqrt{x+sqrt{x+11}}+sqrt{x-sqrt{x+11}})^2 =4^2\ \ (sqrt{x+sqrt{x+11}})^2+2sqrt{x+sqrt{x+11}}sqrt{x-sqrt{x+11}}+(sqrt{x-sqrt{x+11}})^2 =4^2\ \ x+sqrt{x+11}+2sqrt{(x+sqrt{x+11})(x-sqrt{x+11})}+x-sqrt{x+11}=16\ \ 2x+2sqrt{x^2-(sqrt{x+11})^2} =16\ \ 2sqrt{x^2-(x+11)} =16-2x\ \ sqrt{x^2-x-11} =8-x$

$left { {{8-xgeq0} atop {(sqrt{x^2-x-11})^2=(8-x)^2}} right. \ \ left { {{-xgeq-8} atop {x^2-x-11=8^2-2cdot8cdot x+x^2}} right. \ \ left { {{xleq8} atop {x^2-x-11=64-16x+x^2}} right. \ \ left { {{xleq8} atop {15x=75}} right. \ \ left { {{xleq8} atop {x=5}} right. \ \ x=5$

Проверка:

$sqrt{5+sqrt{5+11}}+sqrt{5-sqrt{5+11}} =sqrt{5+sqrt{16}}+sqrt{5-sqrt{16}} =\ =sqrt{5+4}+sqrt{5-4}=sqrt{9}+sqrt{1}=3+1=4$

$4=4$-верно

Ответ: 5