The solve function will solve equations with radicals by using new auxiliary variables and adding new equations.

In some cases the resulting problem will be a system of equations, but in many simple cases radical equations will be handled with transformations and post-processing. For example, the equation $x^2+\sqrt{x+1}=3$ is transformed into $z^4-2z^2+z=2$ using the substitution $x=z^2-1$.

$\mathrm{solve}\left(\mathrm{sqrt}\left(x+1\right)-\mathrm{sqrt}\left(x-1\right)=a\,x\right)$

$\frac{a^4+4}{4a^2}$

$\mathrm{solve}\left(x+\mathrm{sqrt}\left(x\right)-2\,x\right)$

$1$

$\mathrm{solve}\left(x+\mathrm{sqrt}\left(x\right)+x^{\frac{1}{3}}=3\,x\right)$

$1$

$\mathrm{solve}\left(\mathrm{sqrt}\left(x\right)+\mathrm{sqrt}\left(x+1\right)=3\right)$

$\frac{16}{9}$

$\mathrm{solve}\left(\left\{x-y+1\,\mathrm{sqrt}\left(x\right)+\mathrm{sqrt}\left(y\right)-2\right\}\,\left\{x\,y\right\}\right)$

$\left\{x=\frac{9}{16}\,y=\frac{25}{16}\right\}$