Aufgrund von Fakt ist

${\displaystyle {}x_{1}={\frac {{\sqrt {p^{2}-4q}}-p}{2}}\,}$

und

${\displaystyle {}x_{2}={\frac {-{\sqrt {p^{2}-4q}}-p}{2}}\,.}$

Daher ist

${\displaystyle {}{\begin{aligned}x_{1}+x_{2}&={\frac {{\sqrt {p^{2}-4q}}-p}{2}}+{\frac {-{\sqrt {p^{2}-4q}}-p}{2}}\\&={\frac {{\sqrt {p^{2}-4q}}-p-{\sqrt {p^{2}-4q}}-p}{2}}\\&={\frac {-2p}{2}}\\&=-p\end{aligned}}}$

und

${\displaystyle {}{\begin{aligned}x_{1}\cdot x_{2}&={\frac {{\sqrt {p^{2}-4q}}-p}{2}}\cdot {\frac {-{\sqrt {p^{2}-4q}}-p}{2}}\\&={\frac {{\left({\sqrt {p^{2}-4q}}-p\right)}{\left(-{\sqrt {p^{2}-4q}}-p\right)}}{4}}\\&={\frac {-p^{2}+4q+p^{2}}{4}}\\&={\frac {4q}{4}}\\&=q.\end{aligned}}}$