Part 5/7:
When deriving the solutions ( y_1, y_2, ) and ( y_3 ), we incorporate these cube roots of unity to illustrate how the cubic equation can manifest different roots through varying formulations. The general solutions thus become complex due to the presence of imaginary numbers.
The patterns emerge through utilizing a combination of the principal root derived earlier and the square root values influenced by the coefficients of the original equation. Each solution corresponds uniquely to the specific cube roots.