Theory of Equation

Cubic Equations

483 To solve the general cubic equation 𝑥³+𝑝𝑥²+𝑞𝑥+𝑟=0 Remove the term 𝑝𝑥² by the method of (429). Let the transformed equation be 𝑥³+𝑞𝑥+𝑟=0 484 Cardan's method: The complete theoretical solution of this equation by Cardan's method is as follows:- Put 𝑥=𝑦+𝑧i. 𝑦³+𝑧³+(3𝑦𝑧+𝑞)(𝑦+𝑧)+𝑟=0 Put 3𝑦𝑧+𝑞=0; ∴𝑦=−𝑞3𝑧 Substitute this value of 𝑦, and solve the resulting quadratic in 𝑦³. The roots are equal to 𝑦³ and 𝑧³ respectively; and we have, by [i] 485 𝑥=−𝑟2+𝑟²4+𝑞³27¹³+−𝑟2−𝑟²4+𝑞³27¹³ The cubic must have one real root at least, by (409). Let 𝑚 be one of the three values of −𝑟2+𝑟²4+𝑞³27¹³, and 𝑛 one of the three values −𝑟2−𝑟²4+𝑞³27¹³. 486 Let 1, 𝛼, 𝛼² be the three cube roots of unity, so that 𝛼=−12+12-3, and 𝛼²=−12−12-3472 487 Then, since ³𝑚³=𝑚³1, the roots of the cubic will be 𝑚+𝑛, 𝛼𝑚+𝛼²𝑛, 𝛼²𝑚+𝛼𝑛, Now, if in the expansion of −𝑟2±𝑟²4+𝑞³27¹³ by the Binomial Theorem, we put 𝜇= the sum of the odd terms, and 𝜈= the sum of the even terms then we shall have 𝑚=𝜇+𝜈, and 𝑛=𝜇−𝜈; or else 𝑚=𝜇+𝜈−1, and 𝑛=𝜇−𝜈−1; according as 𝑟²4+𝑞³27 is real or imaginary. 488 By substituting these expressions for 𝑚 and 𝑛 in (487), it appears that (i.) If 𝑟²4+𝑞³27 be positive, the roots of the cubic will be 2𝜇, −𝜇+𝜈−3, −𝜇−𝜈−3 (ii.) If 𝑟²4+𝑞³27 be negative, the root will be 2𝜇, −𝜇+𝜈3, −𝜇−𝜈3 (iii.) If 𝑟²4+𝑞³27=0, the roots are 2𝑚, −𝑚, −𝑚 since 𝑚 is now equal to 𝜇. 489 The Trigonometrical method: The equation 𝑥³+𝑝𝑥²+𝑞𝑥+𝑟=0 may be solved in the following manner by Trigonometry, when 𝑟²4+𝑞³27 is negative.
Assume 𝑥=𝑛cos 𝛼. Divide the equation by 𝑛³; thus cos³𝛼+𝑞𝑛²cos 𝛼+𝑟𝑛³=0 But cos³𝛼−34cos 𝛼−cos 3𝛼4=0By (657) Equate coefficients in the two equations; the result is 𝑛=−4𝑞3¹², cos 3𝛼=−4𝑟−34𝑞¹² 𝛼 must now be found with the aid of the Trigonometrical tables. 490 The roots of the cubic will be 𝑛cos 𝛼, 𝑛cos(23𝜋+𝛼), 𝑛cos(23𝜋−𝛼) 491 Observe that, according as 𝑟²4+𝑞³27 is positive or negative, Cardan's method or the Trigonometrical will be practicable. In the former case, there will be one real and two imaginary roots; in the latter case, three real roots.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive