Question

Помогите пожалуйста решить!!!

Answer 1

$sqrt{1+log_2x}+sqrt{4log_4x-2}}=4\\ODZ:; ; left { {{1+log_2xgeq 0} atop {4log_4x-2geq 0}} right. ; left { {{log_2xgeq -1} atop {log_4xgeq frac{1}{2}}} right. ; left { {{xgeq frac{1}{2}} atop {xgeq 2}} right. ; ; to ; ; xgeq 2\\4log_4x=4log_{2^2}x=4cdot frac{1}{2}log_2x=2log_2x\\t=log_2x; ,; ; sqrt{1+t}+sqrt{2t-2}=4\\sqrt{1+t}=4-sqrt{2t-2}\\1+t=16-8sqrt{2t-2}+2t-2\\8sqrt{2t-2}=t+13\\64(2t-2)=t^2+26t+169\\t^2-102t+297=0\\D/4=51^2-297=2304,; ; sqrt{D}=48$

$t_1=51-48=3; ,; ; t_2=51+48=99\\log_2x=3; ; to ; ; x=2^3=8in ODZ\\log_2x=99; ,; ; x=2^{99}$

$Proverka:x=2^{99},sqrt{1+log_22^{99}}+sqrt{4log_42^{99}-2}=4,\\sqrt{1+99}+sqrt{2*99-2}=4\\10+13=4; ; neverno\\x=2^3,sqrt{1+log_22^3}+sqrt{4log_42^3-2}=4\\sqrt{1+3}+sqrt{6-2}=4\\2+2=4\\4=4; ; verno\\Otvet:x=8$