Решите уравнение: (2x+3)/(x+2)=(3x+2)/x.

Ответ:

\[\frac{2x + 3}{x + 2} = \frac{3x + 2}{x}\]

\[ОДЗ:\ \ x \neq - 2\]

\[\ \ \ \ \ \ \ \ \ \ \ x \neq 0\]

\[\frac{2x + 3}{x + 2} - \frac{3x + 2}{x} = 0\]

\[\frac{x(2x + 3) - (x + 2)(3x + 2)}{x(x + 2)} =\]

\[= 0\]

\[\frac{2x^{2} + 3x - \left( 3x^{2} + 2x + 6x + 4 \right)}{x(x + 2)} =\]

\[= 0\]

\[\frac{2x^{2} + 3x - 3x^{2} - 8x - 4}{x(x + 2)} = 0\]

\[\frac{- x^{2} - 5x - 4}{x(x + 2)} = 0\]

\[- x^{2} - 5x - 4 = 0\ \ \ \ | \cdot ( - 1)\]

\[x² + 5x + 4 = 0\]

\[x_{1} + x_{2} = - 5\]

\[x_{1} \cdot x_{2} = 4 \Longrightarrow x_{1} = - 4;\ \ \ \]

\[x_{2} = - 1\]

\[Ответ:\ \ x = - 4\ \ \ \ и\ \ \ x = - 1.\]