$$ax^2+bx+c = 0 \quad(a!=0) \quad\delta>0$$

韦达定理

$$Vieta’s \left{\begin{matrix}
x_1 + x_2 = \frac{-b}{a} \ x_1x_2 = \frac{c}{a}
\end{matrix}\right.$$

用法

1

$$
x_1^2+x_2^2=(x_1+x_2)^2-2x_1x_2
$$

2

$$ |x1 - x2| = \sqrt{(x_1 - x_2)^2} \quad\color{green}=\frac{\sqrt\delta}{|a|}$$
$$ =\sqrt{(x_1-x_2)^2-4x_1x_2} $$
$$ =\sqrt{\frac{b^2}{a^2}-\frac{4c}{a}} $$
$$ =\sqrt{\frac{b^2-4ac}{a^2}} $$