Theory of Equation

Biquadratic Equations

492 Descurles' Solution: To solve the equation 𝑥⁴+𝑞𝑥²+𝑟𝑥+𝑠=0i. the term in 𝑥³ having been removed by the method of (429). Assume (𝑥²+𝑐𝑥+𝑓)(𝑥²−𝑐𝑥+𝑔)=0ii. Multiply out, and equate coefficients with [i.]; and the following equations for determining 𝑓, 𝑔, and 𝑐 are obtained 𝑔+𝑓=𝑞+𝑐², 𝑔−𝑓=𝑟𝑐, 𝑔𝑓=𝑠iii. 493 𝑐⁶+2𝑞𝑐⁴+(𝑞²−4𝑠)𝑐²−𝑟²=0iv. 494 The cubic in 𝑐² is reducible by Cardan's method, when the biquadratic has two real and two imaginary roots. For proof , take 𝛼±𝑖𝛽, and −𝛼±𝛾 as the roots of [i.], since their sum must be zero. Form the sum of each pair for the values of 𝑐 [see [ii.]], and apply the rules in (488) to the cubic in 𝑐².
If the biquadratic has all its roots real, or all imaginary roots of [i.], and form the values of 𝑐 as before. 495 If 𝛼², 𝛽², 𝛾² be the roots of the cubic in 𝑒², the roots of the biquadratic will be −12(𝛼+𝛽+𝛾), 12(𝛼+𝛽−𝛾), 12(𝛽+𝛾−𝛼), 12(𝛾+𝛼−𝛽) For proof, take 𝑤, 𝑥, 𝑦, 𝑧 for the roots of the biquadratic; then, by [ii.], the sum of each pair must give a value of 𝑒. Hence, we have only to solve the symmetrical equations. 𝑦+𝑧=𝛼, 𝑤+𝑥=−𝛼, 𝑧+𝑥=𝛽, 𝑤+𝑦=−𝛽, 𝑥+𝑦=𝛾, 𝑤+𝑧=−𝛾. 496 Ferrari's solution: To the left member of the equation 𝑥⁴+𝑝𝑥³+𝑞𝑥²+𝑟𝑥+𝑠=0 add the quantity 𝑎𝑥²+𝑏𝑥+𝑏²4𝑎, and assume the result =𝑥²+𝑝2𝑥+𝑚² 497 Expanding and equating coefficients, the following cubic equation for determining 𝑚 is obtained 8𝑚³−4𝑞𝑚²+(2𝑝𝑟−8𝑠)𝑚+4𝑞𝑠−𝑝²𝑠−𝑟=0 then 𝑥 is given by the two quadratics 𝑥²+𝑝2𝑥+𝑚=±2𝑎𝑥+𝑏2𝑎 498 The cubic in 𝑚 is reducible by Cardan's method when the biquadratic has two real and two imaginary roots. Assume 𝛼, 𝛽, 𝛾, 𝛿 for the roots of the biquadratic; then 𝛼𝛽 and 𝛾𝛿 are the respective products of roots of the two quadratics above. From this find 𝑚 in terms of 𝛼𝛽𝛾𝛿. 499 Euler's solution: Remove the term in 𝑥³; then we have 𝑥⁴+𝑞𝑥²+𝑟𝑥+𝑠=0 500 Assume 𝑥=𝑦+𝑧+𝑢, and it may be shewn that 𝑦², 𝑧², and 𝑢² are the roots of the equation 𝑡³+𝑞2𝑡²+𝑞²−4𝑠16𝑡−𝑟²64=0 501 The six values of 𝑦, 𝑧, 𝑢, thence obtained, are restricted by the relation 𝑦𝑧𝑢=−𝑟8.
Thus 𝑥=𝑦+𝑧+𝑢 will take four different values.

Sources and References

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