Solution to a quintic:

Numerical values:

Real solutions to an exp-log equation:

Real solution to a system of equations:

Some exact values are generated automatically:

Evaluate to high precision:

Roots of a polynomial:

Real root of a triangular system of equations:

Real roots of an exp-log function:

Roots of an analytic function in a bounded region:

A triple root:

Roots of a polynomial with symbolic coefficients:

Roots of a quadratic with symbolic coefficients:

When a, b, c and the roots are real, the roots are always ordered by their values:

The "standard" formulas for the roots of a quadratic do not guarantee the ordering of roots:

Find the series with respect to a parameter:

Complex components of roots:

Exact comparisons:

Algebraic number coefficients are automatically lifted to integers:

An alternative way of representing the same number:

Find Puiseux series at branch points:

Solve polynomial equations of any degree in closed form in terms of Root:

Solve the characteristic equation of a Hilbert matrix:

Using Eigenvalues:

Find the minimum of a parameterized polynomial:

Solve a constant coefficient differential equation of any degree:

Solve a constant coefficient difference equation of any degree:

Find a solution of a triangular system of equations:

Represent the solution in the Root[f,k] form:

Here, the Root[f,k] representation would involve a polynomial of degree 1000000:

Compute an approximate value of the solution:

Resolve a piecewise function:

Solve univariate exp-log equations and inequalities over the reals:

Solve univariate elementary function equations over bounded intervals and regions:

Solve univariate analytic equations over bounded intervals and regions:

Find real roots of high-degree sparse polynomials and algebraic functions:

Solve univariate transcendental optimization problems:

Integrate a piecewise function with an exp-log inequality condition:

Evaluate the hard hexagon entropy constant:

Solve Kepler's equation:

Compute the Laplace limit constant:

Plot a root as a function of a parameter:

Extract the polynomial from a Root object:

Use ToRadicals to attempt to convert a Root object to radical roots:

This root of a cubic polynomial can be expressed in terms of cube roots:

Series expansions of implicit solutions to equations:

Use RootReduce to canonicalize algebraic numbers, including from operations:

From a triangular system:

Simplify combinations of Root objects:

Reduce an equation for a parameter in a Root object:

Use RootApproximant to generate Root objects from numbers:

Roots are numeric expressions:

Series at branch points may not be valid in all directions:

Canonicalization is only possible for parameter‐free roots:

Parameterized roots can have complicated branch cuts in the complex parameter plane:

A non-polynomial Root object may represent a cluster of distinct roots:

Numerical computation with a higher precision yields an approximation of one of the roots:

The choice of root stays the same for subsequent computations:

A larger setting for $MaxExtraPrecision can be needed for roots with noninteger coefficients:

A high power of a Pisot number that is nearly an integer: