9.1. Find the ratio $\frac{b^{2}}{a c}$ if it is known that one of the roots of the equation $a x^{2}+b x+c=0$ is four times the other $(a c \neq 0)$.
Answer: $\frac{25}{4}$.

Solution: Let the equation have roots $x_{1}, x_{2}\left(x_{1}=4 x_{2}\right)$. Then, from Vieta's formulas, we get: $\quad x_{1}+x_{2}=5 x_{2}=-\frac{b}{a}, \quad x_{1} x_{2}=4 x_{2}^{2}=\frac{c}{a} . \quad$ Hence, $\quad x_{2}=-\frac{b}{5 a}, \quad$ and $4 x_{2}^{2}=\frac{c}{a}=4\left(-\frac{b}{5 a}\right)^{2}=\frac{4 b^{2}}{25 a^{2}}$, which means $\frac{c}{a}=\frac{4 b^{2}}{25 a^{2}}$. Therefore, $c=\frac{4 b^{2}}{25 a}$. Thus, $\frac{b^{2}}{a c}=\frac{25}{4}$.

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