Category:Cardano's Formula

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This category contains pages concerning Cardano's Formula:


Let $P$ be the cubic equation:

$a x^3 + b x^2 + c x + d = 0$ with $a \ne 0$

Then $P$ has solutions:

\(\ds x_1\) \(=\) \(\ds S + T - \dfrac b {3 a}\)
\(\ds x_2\) \(=\) \(\ds -\dfrac {S + T} 2 - \dfrac b {3 a} + \dfrac {i \sqrt 3} 2 \paren {S - T}\)
\(\ds x_3\) \(=\) \(\ds -\dfrac {S + T} 2 - \dfrac b {3 a} - \dfrac {i \sqrt 3} 2 \paren {S - T}\)


where:

\(\ds S\) \(=\) \(\ds \sqrt [3] {R + \sqrt {Q^3 + R^2} }\)
\(\ds T\) \(=\) \(\ds \sqrt [3] {R - \sqrt {Q^3 + R^2} }\)


where:

\(\ds Q\) \(=\) \(\ds \dfrac {3 a c - b^2} {9 a^2}\)
\(\ds R\) \(=\) \(\ds \dfrac {9 a b c - 27 a^2 d - 2 b^3} {54 a^3}\)


Source of Name

This entry was named for Gerolamo Cardano.