Packed into functions like Solve and Reduce are a wealth of sophisticated algorithms, many created specifically for Mathematica. Routinely handling both dense and sparse ...

The main equations that Solve and related Mathematica functions deal with are polynomial equations. It is easy to solve a linear equation in  x. One can also solve quadratic ...

Root[f, k] represents the exact k\[Null]^th root of the polynomial equation f[x] == 0. Root[{f, x_0}] represents the exact root of the general equation f[x] == 0 near x = ...

Numerical root finding. NSolve gives you numerical approximations to all the roots of a polynomial equation. You can also use NSolve to solve sets of simultaneous equations ...

When you give a list of equations to Solve, it assumes that you want all the equations to be satisfied simultaneously. It is also possible to give Solve more complicated ...

The function NDSolve discussed in "Numerical Differential Equations" allows you to find numerical solutions to differential equations. NDSolve handles both single ...

CountRoots[poly, x] gives the number of real roots of the polynomial poly in x.CountRoots[poly, {x, a, b}] gives the number of roots between a and b.

FindRoot[f, {x, x_0}] searches for a numerical root of f, starting from the point x = x_0.FindRoot[lhs == rhs, {x, x_0}] searches for a numerical solution to the equation lhs ...

Although Diophantine equations provide classic examples of undecidability, Mathematica in practice succeeds in solving a remarkably wide range of such equations—automatically ...

NSolve gives you a general way to find numerical approximations to the solutions of polynomial equations. Finding numerical solutions to more general equations, however, can ...