Для каждого значения а решите уравнение: (a^2+5a-24)x=2a^2-5a-3.

Ответ:

\[\left( a^{2} + 5a - 24 \right)x = 2a² - 5a - 3\]

\[a² + 5a - 24 = 0\ \]

\[a_{1} + a_{2} = - 5;\ \ a_{1} \cdot a_{2} = - 24\]

\[\Longrightarrow a_{1} = - 8,\ \ a_{2} = 3.\]

\[a^{2} + 5a - 24 = (a + 8)(a - 3)\]

\[2a^{2} - 5a - 3 = 0\]

\[a_{1} + a_{2} = 2,5;\ \ a_{1} \cdot a_{2} = - 1,5\]

\[\Longrightarrow a_{1} = 3,\ \ a_{2} = - 0,5.\]

\[2a^{2} - 5a - 3 =\]

\[= 2 \cdot (a - 3)(a + 0,5) =\]

\[= (a - 3)(2a + 1)\]

\[(a + 8)(a - 3)x =\]

\[= (a - 3)(2a + 1)\]

\[a = - 8 \Longrightarrow нет\ решения;\]

\[a = - 0,5 \Longrightarrow x = 0;\]

\[a \neq - 8;\ 3;\ - 0,5 \Longrightarrow x = \frac{2a + 1}{a + 8};\]

\[a = 3:\]

\[\left( 3^{2} + 5 \cdot 3 - 24 \right)x =\]

\[= 2 \cdot 3^{2} - 5 \cdot 3 - 3\]

\[0 \cdot x = 0 \Longrightarrow x - любое\ число.\]