Solving Any Cubic

Question

I am trying to show that solving any cubic can be done by intersecting a hyperbola with a parabola. I've tried doing so and substituting, but I continue to get stuck simplifying. I used the hyperbola form $$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$$ and parabola form $$y=a(x-h)^2+c$$ I picked this form since they both had $(x-h)^2$.

Answer 1

If the cubic you're trying to solve is $ax^3+bx^2+cx+d=0$, move the $d$ to the other side and divide by $x$:

$$ax^2+bx+c=\frac{-d}{x}$$

The left-hand side is a parabola, and the right-hand side is a hyperbola.

Another set of rearrangements leads to

$$x^2=-\frac{cx+d}{ax+b}$$

which, again, has a parabola on the left-hand side and a hyperbola on the right-hand side.