Is it always possible to rearrange an equation desirably?

Question

Obviously we can rearrange for $x$ in a polynomial of degree 2.

Let $y=ax^2+bx+c$

then

$x=\frac{-b\pm\sqrt{b^2-4ac+4ay}}{2a}$

Similarly, for $y=ax^3+bx^2+cx+d$, although it is very difficult and long, there apparently also exists a way to make $x$ the subject.

Now I'm wondering whether it is always possible to make $x$ the subject when $y=p_n(x)$, where $p_n(x)$ is any polynomial of degree $n$.

If so, is it always possible to make $x$ the subject when $y=f(x)$, where $f(x)$ is any function of $x$.

And lastly, is there always an exact way to get a desired expression on one side of an equation, obviously still being equivalent to the initial one. If this sounds vague, here is the equation that got me thinking about this:

$y^3+x^3=3xy$

is there a way to make $x$ the subject?

Answer 1

You cannot expect in general to be able to solve for $x$. For instance, consider $$ x^5-4x+2=0. $$ One can easily show, using calculus (or by just plotting) that it has three real roots. One can, however, prove using Galois Theory that no formula exists (that is sums, products, power, roots) that expresses the roots in terms of the coefficients.

And the above is only for polynomial equations. There are many other equations, like trascendental ones, where no closed form solution exists. Example: $x=e^x$.

Answer 2

You have to define what equations you care about and desirably but it seems likely the answer is no. You are presumably familiar with the solutions of linear and quadratic equations. Your equation is a cubic, so you can feed it to Cardano's formula and get $x=$stuff. Quartic equations can be solved, too, but it is such a mess most people ignore the fact. We know there is no general solution for quintic polynomials. Equations which mix exponentials and polynomials also often cannot be solved for one variable. We get asked about them a lot and usually recommend numerical methods to find an approximate solution.