Question

используйте свойства соответствующих функций и решите уравнение ​

Answer 1

$\displaystyle\bf\\\sqrt[3]{x-2} +\sqrt{x+1} =3\\\\\sqrt[3]{x-2} =a \ \ \ \ \Rightarrow \ \ x-2=a^{3} \\\\\sqrt{x+1} =b \ ;\ b\geq 0 \ \ \ \Rightarrow \ \ x+1=b^{2} \\\\\\-\left \{ {{x-2=a^{3} } \atop {x+1=b^{2} }} \right. \\--------\\a^{3} -b^{2} =-3\\\\\\\left \{ {{a+b=3} \atop {a^{3}-b^{2}=-3 }} \right. \\\\\\\left \{ {{b=3-a} \atop {a^{3} -(3-a)^{2}=-3 }} \right. \\\\\\\left \{ {{b=3-a} \atop {a^{3} -9+6a-a^{2} +3=0}} \right.$

$\displaystyle\bf\\\left \{ {{b=3-a} \atop {a^{3} -a^{2}+6a-6=0 }} \right. \\\\\\\left \{ {{b=3-a} \atop {(a^{3} -a^{2})+(6a-6)=0 }} \right. \\\\\\\left \{ {{b=3-a} \atop {a^{2}(a-1)+6(a-1)=0 }} \right. \\\\\\\left \{ {{b=3-a} \atop {(a-1)(a^{2}+6)=0}} \right. \\\\a^{2} +6\neq 0 \ \ , \ \ a-1=0 \ \ \Rightarrow \ \ a=1\\\\\sqrt[3]{x-2} =1\\\\x-2=1\\\\x=3$