Корни уравнения х^2+21х+a=0 относятся как 4 к 3. Найдите корни уравнения и значение a.

\[x^{2} + 21x + a = 0\ \ \]

\[x_{1}\ :x_{2} = 4\ :3 \Longrightarrow x_{1} = \frac{4}{3}x_{2}.\]

\[\left\{ \begin{matrix} x_{1} + x_{2} = - 21\ \ \ \ (1) \\ x_{1} \cdot x_{2} = a\ \ \ \ \ \ \ \ \ \ \ \ (2) \\ \end{matrix} \right.\ \]

\[(1)\text{\ \ }\frac{4}{3}x_{2} + x_{2} = - 21\]

\[\frac{7}{3}x_{2} = - 21\]

\[x_{2} = - 21 \cdot \frac{3}{7}\]

\[x_{2} = - 9.\]

\[(2)\ x_{1} = \frac{4}{3} \cdot ( - 9) = - 12.\]

\[3)\ - 12 \cdot ( - 9) = a\]

\[a = 108.\]

\[Ответ:\ \ x_{1} = - 12;\ \ x_{2} = - 9;\ \ \]

\[a = 108.\]