Question

GPT : \(\sqrt[3]{x+1}+\sqrt[3]{x+2}=1+\sqrt[3]{x^2+3x+2}\)

Answer 1

\(\sqrt[3]{x+1}+\sqrt[3]{x+2}=1+\sqrt[3]{x^2+3x+2}\)

\(\Rightarrow\sqrt[3]{x+1}+\sqrt[3]{x+2}=1+\sqrt[3]{\left(x+1\right)\left(x+2\right)}\)

\(\Rightarrow\sqrt[3]{x+1}-1-\sqrt[3]{x+1}.\sqrt[3]{x+2}+\sqrt[3]{x+2}=0\)

\(\Rightarrow\left(\sqrt[3]{x+1}-1\right)-\sqrt[3]{x+2}\left(\sqrt[3]{x+1}-1\right)=0\)

\(\Rightarrow\left(\sqrt[3]{x+1}-1\right)\left(1-\sqrt[3]{x+2}\right)=0\)

Th1 : \(\sqrt[3]{x+1}-1=0\Rightarrow\sqrt[3]{x+1}=1\)

\(\Rightarrow x+1=1\Rightarrow x=0\)

Th2 : \(\sqrt[3]{x+2}-1=0\Rightarrow\sqrt[3]{x+2}=1\)

\(\Rightarrow x+2=1\Rightarrow x=-1\)

Vậy \(x\in\left\{0;-1\right\}\)