Ta có : \(x^3-y^3-2y^2-3y-1=0\)
\(\Leftrightarrow x^3-\left(y^3+2y^2+3y+1\right)=0\)
\(\Leftrightarrow x^3=y^3+2y^2+3y+1\)
Lại có :
\(y^3+2y^2+3y+1=\left(y^3-3y^2+3y-1\right)+5y^2+2=\left(y-1\right)^3+5y^2+2\)
Do \(5y^2\ge0\forall y\Rightarrow\left(y-1\right)^3+5y^2+2\ge\left(y-1\right)^3+2>\left(y-1\right)^3\left(1\right)\)\(y^3+2y^2+3y+1=\left(y^3+3y^2+3y+1\right)-y^2=\left(y+1\right)^3-y^2\)
Do \(y^2\ge0\forall y\Rightarrow\left(y+1\right)^3-y^2\le\left(y+1\right)^3\forall y\left(2\right)\)
Từ ( 1 ) ; ( 2 )
\(\Rightarrow\left(y-1\right)^3< x^3\le\left(y+1\right)^3\)
\(\Rightarrow\left[{}\begin{matrix}x^3=\left(y+1\right)^3\left(3\right)\\x^3=y^3\left(4\right)\end{matrix}\right.\)
Từ ( 3 )
\(\Rightarrow x^3=y^3+3y^2+3y+1\)
\(\Rightarrow y^3+2y^2+3y+1=y^3+3y^2+3y+1\)
\(\Rightarrow y^2=0\)
\(\Rightarrow y=0\)
\(\Rightarrow\left(y+1\right)^3=1\)
\(\Rightarrow x^3=1\)
\(\Rightarrow x=1\)
Từ ( 4 )
\(\Rightarrow y^3+2y^2+3y+1=y^3\)
\(\Rightarrow2y^2+3y+1=0\)
\(\Rightarrow2y^2+2y+y+1=0\)
\(\Rightarrow2y\left(y+1\right)+y+1=0\)
\(\Rightarrow\left(2y+1\right)\left(y+1\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}2y+1=0\\y+1=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}y=-\dfrac{1}{2}\left(L;y\in Z\right)\\y=-1\end{matrix}\right.\)
\(\Rightarrow y^3=-1=x^3\)
\(\Rightarrow x=-1\)
Vậy \(\left(x,y\right)\in\left\{\left(-1,-1\right);\left(1,0\right)\right\}\)
