Identification of fourth order reciprocal algebraic equation

Question

Problem:

To give the criteria for identification of the fourth order reciprocal algebraic equations.

As provided by E. Galua theory the general algebraic equations for a polynomial of fourth order

ax⁴ +bx³ + cx² + dx + f = 0                                      (*)

is the maximum order type of algebraic equations the solution to which one can write down in radical expressions.

Among all the equations of fourth order (not counting the degenerated cases like  ax⁴ +bx³ = 0, or biquadratic equations ax⁴ + bx² + c = 0) there exist so called reciprocal algebraic equations that can be reduced to the form

a(x + 1 / 2)² + b(x + 1/ x) +c = 0 ,

and thus be solved by repeating procedure of the quadratic equation solution. The question is how to identify the reciprocal equation if we meet it in its canonical form (*).

Provide complete and step by step solution for the question and show calculations and use formulas.