I found that taking a general nonlinear equation with powers of .0 and .5
e.g. $$x^2 + x^{1.5} - x + x^{0.5} + 2 = 0$$
They can be transformed into a polynomial equation by multiplying by the same equation, switching the signs of the coefficients of variables with non integer powers.
e.g. with the above equations, multiply by $$x^2 - x^{1.5} - x - x^{0.5} + 2 = 0$$
I was wondering if this is already a known, and also (if anyone has the time to check) if this would have any practical applications.
Such "algebraization" is always possible for equations with rational powers, using polynomial resultants. In the given case for example, substituting $\,\sqrt{x}=y\,$ then eliminating $\,y\,$ between the resulting equation and $\,y^2-x=0\,$ gives
resultant[ y^4 + y^3 - y^2 + y + 2, y^2 - x, y ]$\,=x^4 - 3 x^3 + 3 x^2 - 5 x + 4\,$. This is the same polynomial obtained via multiplication by the "conjugate" as suggested in the original post or, alternatively, obtained by isolating the square roots on one side then squaring the equation.The technique works for arbitrary rational exponents, not just square roots. Consider for example the more complicated equation:
$$ x+\sqrt{x}+\sqrt[3]{x}-1=0 \tag{1} $$
The substitution $\,\sqrt[6]{x}=y\,$ rewrites the equation as $\,y^6+y^3+y^2-1=0\,$, then eliminating $\,y\,$ between the latter and $\,y^6-x=0\,$ via
resultant[ y^6 + y^3 + y^2 - 1, y^6 - x, y ]gives a polynomial with integer coefficients which has all solutions of $(1)$ as roots:$$ x^6 - 9 x^5 + 32 x^4 - 45 x^3 + 31 x^2 - 11 x + 1 \tag{2} $$
It should be noted that in all cases the resulting equation may (and in general will) have extraneous roots introduced along the way, which are not roots of the original equation. For example, $(2)$ has the root $x=1$ which does not satisfy equation $(1)$.