Editing Cubic equation (section)

===Vieta's substitution===
Vieta's substitution is a method introduced by [[François Viète]] (Vieta is his Latin name) in a text published posthumously in 1615, which provides directly the second formula of {{slink||Cardano's method}}, and avoids the problem of computing two different cube roots.<ref>{{Citation|last = van der Waerden|first = Bartel Leenert|author-link = Bartel Leendert van der Waerden|title = A History of Algebra: From al-Khwārizmī to Emmy Noether|publisher = [[Springer Science+Business Media|Springer-Verlag]]|year = 1985|chapter = From Viète to Descartes|isbn = 3-540-13610-X}}</ref>

Starting from the depressed cubic {{math|''t''<sup>3</sup> + ''pt'' + ''q'' {{=}} 0}}, Vieta's substitution is {{math|''t'' {{=}} ''w'' − {{sfrac|''p''|3''w''}}}}.{{efn|More precisely, Vieta introduced a new variable {{mvar|w}} and imposed the condition {{math|''w''(''t'' + ''w'') {{=}} {{sfrac|''p''|3}}}}. This is equivalent with the substitution {{math|''t'' {{=}} {{sfrac|''p''|3''w''}} − ''w''}}, and differs from the substitution that is used here only by a change of sign of {{mvar|w}}. This change of sign allows getting directly the formulas of {{slink||Cardano's formula}}.}}

The substitution {{math|''t'' {{=}} ''w'' – {{sfrac|''p''|3''w''}}}} transforms the depressed cubic into
<math display="block">w^3+q-\frac{p^3}{27w^3}=0.</math>

Multiplying by {{math|''w''<sup>3</sup>}}, one gets a quadratic equation in {{mvar|w{{sup|3}}}}:
<math display="block">(w^3)^2+q(w^3)-\frac{p^3}{27}=0.</math>

Let 
<math display="block">W=-\frac q 2\pm\sqrt{\frac{p^3}{27} + \frac {q^2} 4}</math> 
be any nonzero root of this quadratic equation. If {{math|''w''<sub>1</sub>}}, {{math|''w''<sub>2</sub>}} and {{math|''w''<sub>3</sub>}} are the three [[cube root]]s of {{mvar|W}}, then the roots of the original depressed cubic are {{math|''w''<sub>1</sub> − {{sfrac|''p''|3''w''<sub>1</sub>}}}}, {{math|''w''<sub>2</sub> − {{sfrac|''p''|3''w''<sub>2</sub>}}}}, and {{math|''w''<sub>3</sub> − {{sfrac|''p''|3''w''<sub>3</sub>}}}}. The other root of the quadratic equation is <math>\textstyle -\frac {p^3}{27W}.</math> This implies that changing the sign of the square root exchanges {{math|''w''<sub>''i''</sub>}} and {{math|− {{sfrac|''p''|3''w''<sub>''i''</sub>}}}} for {{math|1=''i'' = 1, 2, 3}}, and therefore does not change the roots. This method only fails when both roots of the quadratic equation are zero, that is when {{math|''p'' {{=}} ''q'' {{=}} 0}}, in which case the only root of the depressed cubic is {{math|0}}.