# Roots

Roots[lhs==rhs,var]

yields a disjunction of equations which represent the roots of a polynomial equation.

# Details and Options • Roots uses Factor and Decompose in trying to find roots.
• You can find numerical values of the roots by applying N.
• Roots can take the following options:
•  Cubics True whether to generate explicit solutions for cubics EquatedTo Null expression to which the variable solved for should be equated Modulus 0 integer modulus Multiplicity 1 multiplicity in final list of solutions Quartics True whether to generate explicit solutions for quartics Using True subsidiary equations to be solved
• Roots is generated when Solve and related functions cannot produce explicit solutions. Options are often given in such cases.
• Roots gives several identical equations when roots with multiplicity greater than one occur.

# Examples

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## Basic Examples(1)

Find roots of univariate polynomial equations:

## Scope(7)

Equation with exact numeric coefficients:

Equation with symbolic coefficients:

General equations of degree five and higher cannot be solved in radicals:

This equation of degree nine is solved in radicals using factorization and decomposition:

An equation with inexact numeric coefficients:

Multiple roots are repeated the corresponding number of times:

Find roots over the integers modulo 7:

## Options(10)

### Cubics(3)

By default Roots uses the general formulas for solving cubic equations in radicals:

With , Roots does not use the general formulas for solving cubics in radicals:

Solving this cubic equation in radicals does not require the general formulas:

### EquatedTo(1)

Use EquatedTo to specify the left-hand side of the returned equations:

### Modulus(1)

Find roots over the integers modulo 12:

### Multiplicity(1)

With Multiplicity->n, the multiplicity of each root is multiplied by n:

### Quartics(3)

By default Roots uses the general formulas for solving quartic equations in radicals:

With , Roots does not use the general formulas for solving quartics:

Solving this quartic equation in radicals does not require the general formulas:

### Using(1)

Specify equations satisfied by symbolic parameters:

## Properties & Relations(5)

Solutions returned by Roots satisfy the equation:

Use ToRules to convert equations returned by Roots to replacement rules:

Solve uses Roots to find solutions of univariate equations and returns replacement rules:

Roots finds all complex solutions:

Use Reduce to find solutions over specified domains:

Use FindInstance to find one solution:

Use Solve or Reduce to find solutions of systems of multivariate equations:

Use Reduce to find solutions of systems of equations and inequalities:

Use NRoots to find numeric approximations of roots of a univariate equation: