Solving the quartic

Quartic Equation

(1)
\begin{equation} ax^4+bx^3+cx^2+dx+e=0 \end{equation}

To solve the quartic:

(2)
\begin{align} ax^4+bx^3+cx^2+dx+e=0 \\ x^4+\frac{b}{a}x^3+\frac{c}{a}x^2+\frac{d}{a}x+\frac{e}{a}=0 \\ x=u-\frac{b}{4a} \\ \left(u-\frac{b}{4a}\right)^4+\frac{b}{a}\left(u-\frac{b}{4a}\right)^3 \\ +\frac{c}{a}\left(u-\frac{b}{4a}\right)^2+\frac{d}{a}\left(u-\frac{b}{4a}\right) \\ +\frac{e}{a}=0 \\ \left(u^4-\frac{b}{a}u^3+\frac{6b^2}{16a^2}u^2-\frac{4b^3}{64a^3}u+\frac{b^4}{256a^4}\right) \\ +\frac{b}{a}\left(u^3-\frac{3b}{4a}u^2+\frac{3b^2}{16a^2}u-\frac{b^3}{64a^3}\right) \\ +\frac{c}{a}\left(u^2-\frac{b}{2a}u+\frac{b^2}{16a^2}\right)+\frac{d}{a}\left(u-\frac{b}{4a}\right)+\frac{e}{a}=0 \\ u^4+\left(\right)u^2+\left(\right)u+\left(\right)=0 \\ A= \\ B= \\ C= \\ u^4+Au^2+Bu+C=0 \\ \end{align}
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License