Part 3/7:
The challenge lies in the complexity of this formula, which combines both real and complex numbers, depending on the discriminant of the cubic equation.
Derivation of the Cubic Formula
To establish the cubic formula's proof, we begin with a substitution method akin to the completion of squares used in the quadratic case. The cubic equation can be simplified by removing the ( x^2 ) term through a transformation ( x = y - \frac{b}{3a} ), leading to a new function where we can apply the PQ substitution method.
Next, we transform the cubic equation into a simplified form that can be solved as a quadratic equation. This uses a form ( y^3 + py + q = 0 ), which allows us to focus solely on the cubic roots without the complexity of the quadratic term.