We need to find constants .
Thus one solution is,
Note, it fails when
if you did this on your own, then congratulations! But... you are more than 500 years too late. This method is known (for me) as the Cardano formula of the reduced cubic equation. This method was detected first by Scipione del Ferro in 1501 or 1502 and afterwards published by Cardano.
Have a look here: Scipione del Ferro - Wikipedia, the free encyclopedia
Nevertheless you did a great job!