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Forgive me but I don't know the formal term for the type of equation I am talking about, but it is of the form:

$$Ax^n + Bx^2 + Cx + D = 0$$

where $A$, $B$, $C$ and $D$ are constants and $n$ is some decimal number such as $2.3256 \ldots$, etc.

In general is such an expression solvable?

Thanks,

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    There's no analytic ways to solve it. You've to use numerical methods. – 2017-02-28
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    No, however you can use a numerical method such as the [Newton-Raphson Method](https://en.wikipedia.org/wiki/Newton's_method). If you like, I can provide an example solve provided that you give me some values of $A,B,C,D$ and $n$. – 2017-02-28
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    Some of theory, esp. Descartes rule of signs, still applies in locating real roots. But it would be a very special case of non-integer exponent $n$ that allows you to rewrite so that a closed-form solution in radicals is possible. – 2017-02-28

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No, according to Galois theory, only linear, quadratic, cubic and quartic equations can be solved in general.

It is not meant that none of equations excluding the type mentioned above, are solvable. For example, equations like $ax^{2n}+bx^n+c=0 $, or the equation $x^{1/2}-x^2=1$ is solvable.

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    Sometimes it is possible to solve higher degree equations "by radicals" (a phrase you should include in your statement). See some of the Related questions in the sidebar at right (on the desktop version of this page). – 2017-02-28
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If the exponent $n$ can be written ( as reduced fraction) $n=\frac{p}{q}$, then reordering and taking the $q-$power your equation can be writtenas: $A^qx^p=(-Bx^2-Cx-D)^q$.

This is a polynomial equation whose solubility is regutated by the Abel-Ruffini theorem. This means that, in general, we don't have a solution in radical form if the degree of the equation (that is the gratest value from $p$ and $2q$) is $\ge 4$. But for some special values of $A,B,C,D$ some solution can exist that can be espresssed by radicals.

Furthermore, whan we take the $q-$power, if $q$ is an even number, we introduce also the solutions of the equation: $A^qx^q=(Bx^2+Cx+D)^q$, so we have to verify the validity of any algebraic solution.

In conclusion, the best way to solve an equation of the given form, is to use some numerical method as suggested in the comments.