In my research I came across the following derivation of the cubic formula that was almost complete. However at the end it says to make the substitution $u=z^3$ which would determine the roots. I am not quite sure how to do this and then how to determine the roots and I am looking for some help.
A general form for the cubic equation is,
$$ax^3+bx^2+cx+d=0 \tag{1}$$
To find the roots of this equation we first try to get rid of the quadratic term $x^2$. The substitution $x=y-\dfrac{b}{3a}$ helps in achieving our goal. This results in, $$ay^3+\left(c-\dfrac{b^2}{3a}\right)y+\left(d+\dfrac{2b^3}{27a^2}-\dfrac{bc}{3a}\right)=0\tag{2}$$ which we transform into the following, $$y^3+\dfrac{1}{a}\left(c-\dfrac{b^2}{3a}\right)y+\dfrac{1}{a}\left(d+\dfrac{2b^3}{27a^2}-\dfrac{bc}{3a}\right)=0\tag{3}$$ Upon assuming $e=\dfrac{1}{a}\left(c-\dfrac{b^2}{3a}\right)$ and $f=\dfrac{1}{a}\left(d+\dfrac{2b^3}{27a^2}-\dfrac{bc}{3a}\right)$ we get the equation as, $$y^3+ey+f=0\tag{4}$$ We reduce this equation by the substitution $y=z+\dfrac{s}{z}$ and choosing $s=-\dfrac{e}{3}$ we obtain the simplified equation as, $$z^6+fz^3-\dfrac{e^3}{27}=0\tag{5}$$ What only remains is to make the substitution $u=z^3$.
Best Answer
We are free to select either $z_0$ or $z_1$ and we choose $z_0$. Since $z^3=u$ we know there are three solutions \begin{align*} z_0,z_0 e^{\frac{2\pi i}{3}},z_0 e^{\frac{4\pi i}{3}} \tag{3} \end{align*}
[2017-11-05] Add-on: Let's make an example to see the formulas in action.
Find the solutions of \begin{align*} \color{blue}{x^3-7x^2+14x-8=0} \end{align*}
Note: Working with the conjugate complex $u_2=\overline{u_1}=\frac{10}{27}-\frac{i}{\sqrt{3}}$ instead of $u_1$ gives the complex conjugates $\overline{z_1},\overline{z_1}e^{-\frac{2\pi i}{3}}$ and $\overline{z_1}e^{-\frac{4\pi i}{3}}$ and leads (of course) to the same results $x_0=4,x_1=1$ and $x_2=2$.