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Vieta's Substitution

The substitution of

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

into the standard form Cubic Equation
\end{displaymath} (2)

The result reduces the cubic to the equation
w^3-{p^3\over 27w^3}-q=0,
\end{displaymath} (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.
\end{displaymath} (4)

See also Cubic Equation,Quadratic Equation

© 1996-9 Eric W. Weisstein