Tìm các số nguyên x,y thỏa mãn : ( x - 2 ) ^2 . ( y-3 ) = -4

Đáp án: $\left( {x;y} \right) \in \left\{ {\left( {4;0} \right);\left( {0;0} \right);\left( {3; - 3} \right);\left( {1; - 3} \right)} \right\}$

Giải thích các bước giải:

$\begin{array}{l}
{\left( {x - 2} \right)^2}.\left( {y - 3} \right) =  - 4 = 4.\left( { - 1} \right) = 1.\left( { - 4} \right)\\
Do:\left\{ \begin{array}{l}
{\left( {x - 2} \right)^2} \ge 0\\
x \in Z\\
y \in Z
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
{\left( {x - 2} \right)^2} \in N\\
y - 3 \in Z
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
{\left( {x - 2} \right)^2} = 4\\
y - 1 =  - 1
\end{array} \right.\\
\left\{ \begin{array}{l}
{\left( {x - 2} \right)^2} = 1\\
y - 1 =  - 4
\end{array} \right.
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
\left[ \begin{array}{l}
x - 2 = 2\\
x - 2 =  - 2
\end{array} \right.\\
y = 0
\end{array} \right.\\
\left\{ \begin{array}{l}
\left[ \begin{array}{l}
x - 2 = 1\\
x - 2 =  - 1
\end{array} \right.\\
y =  - 3
\end{array} \right.
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
x = 4;y = 0\\
x = 0;y = 0\\
x = 3;y =  - 3\\
x = 1;y =  - 3
\end{array} \right.\\
Vậy\,\left( {x;y} \right) \in \left\{ {\left( {4;0} \right);\left( {0;0} \right);\left( {3; - 3} \right);\left( {1; - 3} \right)} \right\}
\end{array}$