# Problem of solving the cubic function

by Martin Zhao
Tags: cubic, function, solving
 Sci Advisor P: 1,741 For the cubic equation $ax^3+bx^2+cx+d=0$ (in your case the constant term is $d-y$, not $d$), try substituting $x = z +\frac{\gamma}{z}$, and solve for $z$ by choosing the constant $\gamma$ correctly. If fairly certain that for a good choice of $\gamma$ (it will become apparant what $\gamma$ must be) you will end up with a quadratic function in $z^2$. This way you may arrive at the formula yourself, it's a neat exercise. You probably need to be careful verifying your solution afterwards, as $z +\frac{\gamma}{z}$ is not defined everywhere, and does not attain all values. To make calculations easier, you can assume $a = 1$ first, and make the necessary modification afterwards.