The fsolve command numerically computes the zeroes of one or more equations, expressions or procedures.

The solutions to a single equation are returned as an expression sequence.

The solutions to a set or list of equations are returned as sets of equation sequences, if the variables are not specified. If the variables are specified either as a set, list, or Vector, then the solutions are returned in the same form. If the variables are specified as a Vector then the solutions contain only the ordered numeric values, rather than equations for each variable name.

For a single polynomial equation of one variable with real coefficients, by default the fsolve command computes all real (non-complex) roots. It may not return all roots for exceptionally ill-conditioned polynomials.

For a single polynomial equation of one variable with some (non-real) complex coefficients, the fsolve command computes all real and complex roots. It may not return all roots for exceptionally ill-conditioned polynomials.

For a general equation or system of equations, the fsolve command computes a single real root.

For a univariate real polynomial equation, the fsolve command computes all real solutions.

$\mathrm{polynomial}≔2x^5-11x^4-7x^3+12x^2-4x=0\:$

$\mathrm{fsolve}\left(\mathrm{polynomial}\right)$

$\mathrm{-1.334383488},0.,5.929222024$

For more complicated equations, the fsolve command computes one real solution.

$\mathrm{polynomials}≔\left\{3x^4{y}^2=17\,x^{3}y-5x{y}^2-2y=1\right\}\:$

$\mathrm{fsolve}\left(\mathrm{polynomials}\right)$

$\left\{x=2.118203038\,y=0.5305528603\right\}$

$\mathrm{fsolve}\left(\tan \left(\sin \left(x\right)\right)=1\right)$

$0.9033391108$

$f≔\sin \left(x+y\right)-\exp \left(x\right)y=0\:$

$g≔x^2-y=2\:$

$\mathrm{fsolve}\left(\left\{f\,g\right\}\right)$

$\left\{x=\mathrm{-6.017327250}\,y=34.20822723\right\}$

If an equation has no real solutions or you are interested in complex solutions then you can search for a complex solution using the complex option. For equations and expressions, the complex option must be preceded by the variables option.

$\mathrm{fsolve}\left(\ln \left(v\right)=v^2\right)$

$\mathrm{fsolve}\left(\ln \left(v\right)=v^2\,v\right)$

$\mathrm{fsolve}\left(\ln \left(v\right)=v^2\,v\,\mathrm{complex}\right)$

$0.6143632454-0.6810654878\mathrm{I}$

$\mathrm{fsolve}\left(\ln \left(v\right)-v^2\,v\,\mathrm{complex}\right)$

$0.6143632454-0.6810654878\mathrm{I}$

$\mathrm{fsolve}\left(t\mapsto \ln \left(t\right)-t^2\,\mathrm{complex}\right)$

$0.6143632454-0.6810654878\mathrm{I}$