Solving the cubic

Cubic equation:

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

To solve the cubic:

(2)
\begin{align} ax^3+bx^2+cx+d=0 \\ x^3+\frac{b}{a}x^2+\frac{c}{a}x+\frac{d}{a}=0 \\ x=u-\frac{b}{3a} \\ \left(u-\frac{b}{3a}\right)^3+\frac{b}{a}\left(u-\frac{b}{3a}\right)^2+\frac{c}{a}\left(u-\frac{b}{3a}\right)+\frac{d}{a}=0 \\ \left(u^3-\frac{b}{a}u^2+\frac{b^2}{3a^2}u-\frac{b^3}{27a^3}\right)+\frac{b}{a}\left(u^2-\frac{2b}{3a}u+\frac{b^2}{9a^2}\right)+\frac{c}{a}\left(u-\frac{b}{3a}\right)+\frac{d}{a}=0 \\ u^3+\frac{b^2}{3a^2}u-\frac{b^3}{27a^3}-\frac{2b^2}{3a^2}u+\frac{b^3}{9a^3}+\frac{c}{a}u-\frac{bc}{3a^2}+\frac{d}{a}=0 \\ u^3+\frac{3ac-b^2}{3a^2}u+\frac{2b^2-9abc+27a^2d}{27a^3}=0 \\ p=\frac{3ac-b^2}{3a^2} \\ q=\frac{2b^2-9abc+27a^2d}{27a^3} \\ u^3+pu+q=0 \\ \end{align}
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