[latex] \sqrt[3]{x+3}+ \sqrt{5-x}=2[/latex]

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