Для каждого значения а решите уравнение: x^2-(7a+2)x+14a=0.

Ответ:

\[x² - (7a + 2)x + 14a = 0\]

\[D = (7a + 2)^{2} - 4 \cdot 1 \cdot 14a =\]

\[= 49a^{2} + 28a + 4 - 56a =\]

\[= 49a^{2} - 28a + 4 = (7a - 2)^{2}\]

\[При\ D = 0:\]

\[(7a - 2)^{2} = 0\]

\[7a - 2 = 0\]

\[7a = 2\]

\[a = \frac{2}{7}.\]

\[1)\ \ a = \frac{2}{7}:\]

\[x = \frac{7 \cdot \frac{2}{7} + 2 \pm \left| 7 \cdot \frac{2}{7} - 2 \right|}{2} =\]

\[= \frac{4}{2} = 2.\]

\[2)\ D > 0;\ \ a > \frac{2}{7}:\]

\[x_{1} = \frac{7a + 2 + 7a - 2}{2} =\]

\[= \frac{14a}{2} = 7a;\]

\[x_{2} = \frac{7a + 2 - 7a + 2}{2} = \frac{4}{2} = 2.\]

\[3)\ D < 0 \Longrightarrow корней\ нет.\]

\[Ответ:a = \frac{2}{7} \Longrightarrow x = 2;\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ a > \frac{2}{7} \Longrightarrow x = 2\ или\ \]

\[x = 7a.\]