223. Find the dependence between the coefficients of the equation $a x^{2}+b x+c=0$, if the ratio of the roots is 2.
Solution. Let one root of the equation $a x^{2}+b x+c=0$ be $x_{1}$, then the other should be $2 x_{1}$. Therefore,

$$
x_{1}+2 x_{1}=-\frac{b}{a}
$$

or

$$
\begin{aligned}
& 3 x_{1}=-\frac{b}{a} \\
& 2 x_{1} \cdot x_{1}=\frac{c}{a}
\end{aligned}
$$

or

$$
2 x_{1}^{2}=\frac{c}{a}
$$

From equations (1) and (2), we obtain the desired relationship: $2 b^{2}=9 a c$.