Решите уравнение, используя введение новой переменной: (x^2+2x)^2-2*(x^2+2x)-3=0.

Ответ:

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

\[= 0\]

\[Пусть\ \left( x^{2} + 2x \right) = a:\]

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

\[D = 1 + 3 = 4\]

\[a_{1} = 1 + 2 = 3;\ \ \ \]

\[a_{2} = 1 - 2 = - 1.\]

\[Подставим:\]

\[1)\ x^{2} + 2x = 3\]

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

\[D = 1 + 3 = 4\]

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

\[x_{2} = - 1 - 2 = - 3.\]

\[в)\ x^{2} + 2x = - 1\]

\[x^{2} + 2x + 1 = 0\]

\[(x + 1)^{2} = 0\]

\[x = - 1.\]

\[Ответ:\ \ x = \pm 1;\ \ x = - 3.\]