# 希皮奥内·德尔·费罗

## 费罗的一元三次方程解法

${\displaystyle x^{3}+ax^{2}+bx+c=0\,}$

${\displaystyle x^{3}+mx=n\,}$
${\displaystyle x^{3}=mx+n\,}$
${\displaystyle x^{3}+n=mx\,}$
${\displaystyle x^{3}+mx+n=0\,}$

 ${\displaystyle x^{3}+m*x-n=0\,}$ ${\displaystyle x^{3}+6x-20=0\,}$ ${\displaystyle D={\Big (}{\frac {m}{3}}{\Big )}^{3}+{\Big (}{\frac {n}{2}}{\Big )}^{2}}$ ${\displaystyle D={\big (}2{\big )}^{3}+{\big (}10{\big )}^{2}=108}$ ${\displaystyle {\text{ 在 }}D>0{\text{ 的情况下可解}}}$ ${\displaystyle v={\sqrt[{3}]{{\frac {n}{2}}+{\sqrt {D}}}}}$ ${\displaystyle v={\sqrt[{3}]{10+{\sqrt {108}}}}=2.732051}$ ${\displaystyle u={\sqrt[{3}]{{\frac {n}{2}}-{\sqrt {D}}}}}$ ${\displaystyle u={\sqrt[{3}]{10-{\sqrt {108}}}}=-0.732051}$ ${\displaystyle {\text{ 则得到解 }}x=u+v}$ ${\displaystyle \quad x=2.732051-0.732051=2}$ ${\displaystyle {\text{将解代入方程：}}}$ ${\displaystyle \quad 2^{3}+6*2-20=0}$

## 参考文献

1. ^ Cubic Formula （英文）. [2007-06-28]. （原始内容存档于2011-05-22）.