Question

$sqrt[3]{x+3}+ sqrt{5-x}=2$

Answer 1

$sqrt[3]{x+3}+sqrt{5-x} =2; ,; ; ; ODZ:; ; x leq 5\\Zamena:; ; u= sqrt[3]{x+3}; ,; ; v=sqrt{5-x} geq 0; ; Rightarrow \\u^3+v^2=(x+3)+(5-x)=8\\ left { {{u+v=2} atop {u^3+v^2=8}} right. ; left { {{v=2-u} atop {u^3+4-4u+u^2=8}} right. \\u^3+u^2-4u-4=0\\u^2(u+1)-4(u+1)=0\\(u+1)(u^2-4)=0\\(u+1)(u-2)(u+2)=0\\u_1=-1,; u_2=-2,; u_3=2\\v geq 0:; ; v_1=2-u_1=3,; ; v_2=4,; ; v_3=0$

$1); ; u=sqrt[3]{x+3}=-1 ; ,; x+3=-1,; ; x=-4\\v=sqrt{5-x}=3; ,; 5-x=9,; ; x=-4$

$2); ; sqrt[3]{x+3}=-2; ,; ; x+3=-8; ,; x=-11\\sqrt{5-x}=4; ,; ; 5-x=16; ,; ; x=-11 \\3); ; sqrt[3]{x+3}=2; ,; ; x+3=8; ,; ; x=5\\sqrt{5-x}=0; ,; ; 5-x=0; ,; ; x=5\\Otvet:; ; x=-11,; ; x=-4,; ; x=5; .$