3. If the quadratic equation with real coefficients $a x^{2}+b x+c=0$ has 2 imaginary roots $x_{1}, x_{2}$, and $x_{1}^{3} \in R$, then $\frac{a c}{b^{2}}=$
Answer: 1
Explanation: Note that $x_{2}=\overline{x_{1}}$, from $x_{1}^{3} \in R \Rightarrow x_{1}^{3}=\overline{x_{1}^{3}}=\left(\overline{x_{1}}\right)^{3}=x_{2}^{3} \Rightarrow\left(x_{1}-x_{2}\right)\left(x_{1}^{2}+x_{1} x_{2}+x_{2}^{2}\right)=0$
$$
\Rightarrow x_{1}^{2}+x_{1} x_{2}+x_{2}^{2}=0 \Rightarrow\left(x_{1}+x_{2}\right)^{2}=x_{1} x_{2} \Rightarrow\left(-\frac{b}{a}\right)^{2}=\frac{c}{a} \Rightarrow b^{2}=a c \Rightarrow \frac{a c}{b^{2}}=1
$$