224. In the equation $\left(k^{2}-5 k+3\right) x^{2}+(3 k-1) x+2=0$, find the value of $k$ for which the ratio of the roots of the equation would be 2.
Solution. In the equation $a x^{2}+b x+c=0$, the ratio of the roots is 2, which means there is a relationship between the coefficients of the equation: $2 b^{2}=9 a c$ (see problem 223). For the given equation, this relationship will take the form:

$$
2(3 k-1)^{2}=9\left(k^{2}-5 k+3\right) \cdot 2
$$

or

$$
(3 k-1)^{2}=9\left(k^{2}-5 k+3\right)
$$

Solving equation (1), we get that $k=\frac{2}{3}$. We could also have arrived at the same equation without using the conclusion from problem 223, but by using the properties of the roots of a quadratic equation and the condition of the problem.