$$x^2+bx+c=0$$

$$x_1+x_2=-b,x_1x_2=c.$$

$$\begin{cases} x_1+x_2=-b,\\ \alpha_1x_1+\alpha_2x_2=k, \end{cases}$$

$$(\alpha_1x_1+\alpha_2x_2)(\alpha_2x_1+\alpha_1x_2)$$

$$(\alpha_1x_1+\alpha_2x_2)(\alpha_2x_1+\alpha_1x_2)$$

$$(\alpha_1x_1+\alpha_2x_2)(\alpha_2x_1+\alpha_1x_2)=\alpha_1\alpha_2(b^2-2c)+(\alpha_1^2+\alpha_2^2)c.$$

$$\alpha_1x_1+\alpha_2x_2=p(\alpha_2x_1+\alpha_1x_2),$$

$$-\alpha_1^2(x_1-x_2)^2=-\alpha_1^2b^2+4\alpha_1^2c.$$

$$x_1-x_2=\pm\sqrt{b^2-4c}.$$

$$\begin{cases} x_1+x_2=-b,\\ x_1-x_2=\pm \sqrt{b^2-4c} \end{cases}$$

$$x^3+bx^2+cx+d=0.$$

$$\begin{cases} x_1+x_2+x_3=-b,\\ x_1x_2+x_2x_3+x_3x_1=c\\ x_1x_2x_3=-d. \end{cases}$$

$$\begin{cases} \alpha_1x_1+\alpha_2x_2+\alpha_3x_3=k_1,\\ \beta_1x_1+\beta_2x_2+\beta_3x_3=k_2.\\ \end{cases}$$

\label{eq:1}
(\alpha_1x_1+\alpha_2x_2+\alpha_3x_3)(\alpha_1x_1+\alpha_3x_2+\alpha_2x_3)(\alpha_3x_1+\alpha_2x_2+\alpha_1x_3)(\alpha_2x_1+\alpha_1x_2+\alpha_3x_3)(\alpha_3x_1+\alpha_1x_2+\alpha_2x_3)(\alpha_2x_1+\alpha_3x_2+\alpha_1x_3)

• $\alpha_1x_1+\alpha_2x_2+\alpha_3x_3$
• $\alpha_3x_1+\alpha_1x_2+\alpha_2x_3$
• $\alpha_2x_1+\alpha_3x_2+\alpha_1x_3$

• $\alpha_3x_1+\alpha_2x_2+\alpha_1x_3$
• $\alpha_1x_1+\alpha_3x_2+\alpha_2x_3$
• $\alpha_2x_1+\alpha_1x_2+\alpha_3x_3$

$$\alpha_1x_1+\alpha_2x_2+\alpha_3x_3=p(\alpha_3x_1+\alpha_1x_2+\alpha_2x_3)=p^2(\alpha_2x_1+\alpha_3x_2+\alpha_1x_3).$$

$$\alpha_3x_1+\alpha_2x_2+\alpha_1x_3=q(\alpha_1x_1+\alpha_3x_2+\alpha_2x_3)=q^2(\alpha_2x_1+\alpha_1x_2+\alpha_3x_3).$$

$$\alpha_1=\omega_3\alpha_3=\omega_3^2\alpha_2,$$

$$\alpha_1=\omega_3^{-1}\alpha_3=\omega_3^{-2}\alpha_2,$$

$$U^3V^3=f(b,c,d).$$

\begin{align*}
&(\alpha_1x_1+\alpha_2x_2+\alpha_3x_3)(\alpha_3x_1+\alpha_1x_2+\alpha_2x_3)(\alpha_2x_1+\alpha_3x_2+\alpha_1x_3)\\&+(\alpha_3x_1+\alpha_2x_2+\alpha_1x_3)(\alpha_1x_1+\alpha_3x_2+\alpha_2x_3)(\alpha_2x_1+\alpha_1x_2+\alpha_3x_3)=U^3+V^3
\end{align*}

$$\begin{cases} U^3V^3=f(b,c,d),\\ U^3+V^3=g(b,c,d). \end{cases}$$

$$\begin{cases} x_1+x_2+x_3=-b,\\ \alpha_1x_1+\alpha_2x_2+\alpha_3x_3=U,\\ \alpha_3x_1+\alpha_2x_2+\alpha_3x_1=V. \end{cases}$$