Vieta's Substitution

The substitution of

$$x=w-{p\over 3w}$$ (1)

into the standard form Cubic Equation

$$x^3+px=q.$$ (2)

The result reduces the cubic to the equation

$$w^3-{p^3\over 27w^3}-q=0,$$ (3)

which is easily turned into a Quadratic Equation in $w^3$ by multiplying through by $w^3$ to obtain

$$(w^3)^2-q(w^3)-{\textstyle{1\over 27}}p^3=0.$$ (4)

See also Cubic Equation,Quadratic Equation