Question

Помогите решить два иррационального уравнения вообще не шарю

Answer 1

$1)quad sqrt{5+sqrt{x-4}}=3; ,; ; ; ODZ:; left { {{x geq 4} atop {5+sqrt{x-4} geq 0}} right. ; to ; x geq 4\\5+sqrt{x-4}=9\\sqrt{x-4}=4\\x-4=16\\x=20\\Proverka;; ; sqrt{5+sqrt{20-4}}=3; ,; ; sqrt{5+4}=3; ,; 3=3\\Otvet:; ; x=20; .$

$2)quad sqrt{2-sqrt{2+x}}=x; ,\\ ODZ:; left { {{x geq 0,; x+2 geq 0} atop {2-sqrt{x+2} geq 0}} right. ; left { {{x geq 0} atop {sqrt{x+2}leq 2}} right. ; left { {{x geq 0} atop {x+2 leq 4}} right. ; left { {{x geq 0} atop {x leq 2}} right. ; ; ; 0 leq x leq 2\\2-sqrt{2+x}=x^2\\sqrt{2+x}=2-x^2\\2+x=(2-x^2)^2\\2+x=4-4x^2+x^4\\x^4-4x^2-x+2=0\\(x^4+x^3)-x^3-x^2-3x^2-x+2=0\\x^3(x+1)-x^2(x+1)-(3x^2+x-2)=0\\x^3(x+1)-x^2(x+1)-3(x+1)(x-frac{2}{3})=0$

$(x+1)(x^3-x^2-3x+2)=0\\x_1=-1notin ODZ\\x^3-x^2-3x+2=0\\x^2(x-2)+x(x-2)-(x-2)=0\\(x-2)(x^2+x-1)=0; ; to ; ; x_2=2in ODZ; ,\\x^2+x-1=0\\D=1+5=6; ,; ; x_{3,4}=frac{-1pm sqrt6}{2}= left [ {{approx -1,72notin ODZ} atop {approx 0,72in ODZ}} right. \\Otvet:; ; x=2; ,; x=frac{-1+sqrt6}{2}; .$