Решите уравнение, используя введение новой переменной: (x^2+x)^2-5*(x^2+x)+6=0.

Ответ:

\[\left( x^{2} + x \right)^{2} - 5 \cdot \left( x^{2} + x \right) + 6 = 0\]

\[Пусть\ a = x^{2} + x:\]

\[a^{2} - 5a + 6 = 0\]

\[a_{1} + a_{2} = 5;\ \ \ \ a_{1} \cdot a_{2} = 6\]

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

\[Подставим:\]

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

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

\[D = 1 + 12 = 13\]

\[x_{1,2} = \frac{1 \pm \sqrt{13}}{2}.\]

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

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

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

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

\[Ответ:\ \ x = - 2;\ \ x = 1;\ \ \]

\[x = \frac{1 \pm \sqrt{13}}{2}.\]