關於一元三次方程求根公式的解法

Cardano's method
The solutions can be found with the following method due to Scipione del Ferro and Tartaglia, published by Gerolamo Cardano in 1545.
We first divide the given equation by α3 to arrive at an equation of the form


圖片參考：http://upload.wikimedia.org/math/b/e/4/be445df476def6f2c69017c9602cae5d.png

The substitution x = t - a/3 eliminates the quadratic term; in fact, we get the equation


圖片參考：http://upload.wikimedia.org/math/d/e/f/defd9995767bca53697f32844cc55e67.png

This is called the depressed cubic.
Suppose that we can find numbers u and v such that


圖片參考：http://upload.wikimedia.org/math/0/0/a/00ad6a5c23cd8280c0abfa01a1d47ee3.png

A solution to our equation is then given by


圖片參考：http://upload.wikimedia.org/math/d/e/e/dee06fb77038b1e1ffdfe840180f0fa9.png

as can be checked by directly substituting this value for t in (2), as a consequence of the third order binomial identity


圖片參考：http://upload.wikimedia.org/math/d/2/f/d2fa7723056e5b5ed250288b50fb00e9.png

The system (3) can be solved by solving the second equation for v, which gives


圖片參考：http://upload.wikimedia.org/math/4/e/a/4ea7b6c7b6e198c93280c7cc80a439ad.png

Substituting this in the first equation in (3) yields


圖片參考：http://upload.wikimedia.org/math/2/4/4/244110a0c6239bfe1602ddb6ec4c5a8c.png

This can be seen as a quadratic equation for u3. If we solve this equation, we find that


圖片參考：http://upload.wikimedia.org/math/3/7/4/3749839363242264e1aed01e5bb45b77.png

Since t = v − u and t = x + a/3, we find


圖片參考：http://upload.wikimedia.org/math/4/2/8/4283e1db92325eafdc4c47752d1af11c.png

Note that there are six possibilities in computing u with (4), since there are two solutions to the square root (
圖片參考：http://upload.wikimedia.org/math/2/0/8/208fbdaecd37cbf2f90b38416e812355.png
. Second, if p = q = 0, then we have the triple real root x = −a/3.