Đáp án:
x=-1
Giải thích các bước giải:
\(\begin{array}{l}
P = \dfrac{{{x^2} - 4x - 2}}{{{x^2} - 4}} = \dfrac{{{x^2} - 4x + 4 - 6}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}\\
= \dfrac{{{{\left( {x - 2} \right)}^2} - 6}}{{\left( {x - 2} \right)\left( {x + 2} \right)}}\\
= \dfrac{{x - 2}}{{x + 2}} - \dfrac{6}{{{x^2} - 4}}\\
= \dfrac{{x + 2 - 4}}{{x + 2}} - \dfrac{6}{{{x^2} - 4}}\\
= 1 - \dfrac{4}{{x + 2}} - \dfrac{6}{{{x^2} - 4}}\\
P \in Z\\
\Leftrightarrow \left\{ \begin{array}{l}
\dfrac{4}{{x + 2}} \in Z\\
\dfrac{6}{{{x^2} - 4}} \in Z
\end{array} \right.\\
\to \left\{ \begin{array}{l}
x + 2 \in U\left( 4 \right)\\
{x^2} - 4 \in U\left( 6 \right)
\end{array} \right.\\
\to \left\{ \begin{array}{l}
\left[ \begin{array}{l}
x + 2 = 4\\
x + 2 = - 4\\
x + 2 = 2\\
x + 2 = - 2\\
x + 2 = 1\\
x + 2 = - 1
\end{array} \right.\\
\left[ \begin{array}{l}
{x^2} - 4 = 6\\
{x^2} - 4 = - 6\left( l \right)\\
{x^2} - 4 = 3\\
{x^2} - 4 = - 3\\
{x^2} - 4 = 2\\
{x^2} - 4 = - 2\\
{x^2} - 4 = 1\\
{x^2} - 4 = - 1
\end{array} \right.
\end{array} \right. \to \left\{ \begin{array}{l}
\left[ \begin{array}{l}
x = 2\\
x = - 6\\
x = 0\\
x = - 4\\
x = - 1\\
x = - 3
\end{array} \right.\\
\left[ \begin{array}{l}
x = \sqrt {10} \\
x = - \sqrt {10} \\
x = \sqrt 7 \\
x = - \sqrt 7 \\
x = 1\\
x = - 1\\
x = \sqrt 6 \\
x = - \sqrt 6 \\
x = \sqrt 2 \\
x = - \sqrt 2 \\
x = \sqrt 5 \\
x = - \sqrt 5 \\
x = \sqrt 3 \\
x = - \sqrt 3
\end{array} \right.
\end{array} \right.\\
\to x = - 1
\end{array}\)
