Category:Cardano's Formula

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.

The following 13 pages are in this category, out of 13 total.