Generalizing Ruler and Compass

Question

Using Ruler and Compass we can construct any solution to a quadratic equation.

I am wondering whether the idea can be generalized to construct solutions to higher order polynomials.

For example can quadratic curves and compass construct any solution to a cubic, cubic curves and compass construct any solution to a quartic, and so on?

Alternatively can one use a ruler and some alternative transcendental curve to solve higher order polynomials?

Answer 1

Using conics (which are quadratic curves), we can solve any cubic and quartic equation with rational coefficients.