Solve

Solve [ expr , vars ] attempts to solve the system expr of equations or inequalities for the variables vars .

Solve [ expr , vars , dom ] solves over the domain dom . Common choices of dom are Reals , Integers , and Complexes .

Details and Options

The system expr can be any logical combination of:

Examples

Basic Examples (5)

Solve a quadratic equation:

Solve simultaneous equations in and :

Solve an equation over the reals:

Solve an equation over the positive integers:

Solve equations in a geometric region:

Scope (85)

Basic Uses (6)

Solutions are given as lists of replacements:

Use ReplaceAll ( /. ) to replace by solutions:

Check that solutions satisfy the equations:

Solve uses < >to represent the empty solution or no solution:

Solve uses < < >> to represent the universal solution or all points satisfying the equations:

Find solutions over specified domains:

Solve equations with coefficients involving a symbolic parameter:

Plot the real parts of the solutions for y as a function of the parameter a :

Solution of this equation over the reals requires conditions on the parameters:

Replace x by solutions and simplify the results:

Use Normal to remove the conditions:

Solution of this equation over the positive integers requires introduction of a new parameter:

List the first 10 solutions:

Complex Equations in One Variable (15)

Polynomial equations solvable in radicals:

To use general formulas for solving cubic equations, set Cubics  True :

By default, Solve uses Root objects to represent solutions of general cubic equations with numeric coefficients:

General polynomial equations:

Polynomial equations with multiple roots:

Find five roots of a polynomial of a high degree:

Polynomial equations with symbolic coefficients:

Solve cannot find all solutions here:

Find three solutions:

Univariate elementary function equations over bounded regions:

Univariate holomorphic function equations over bounded regions:

Here Solve finds some solutions but is not able to prove there are no other solutions:

Equation with a purely imaginary period over a vertical stripe in the complex plane:

Find a specified number of roots of an unrestricted complex equation:

Nonanalytic complex equations:

Systems of Complex Equations in Several Variables (12)

Systems of linear equations:

Linear equations with symbolic coefficients:

Underdetermined systems of linear equations:

Linear equations with no solutions:

Systems of polynomial equations:

Find five out of a trillion roots of a polynomial system:

Polynomial equations with symbolic coefficients:

Find a specified number of solutions of transcendental equations:

Square analytic systems over bounded boxes:

Real Equations in One Variable (13)

Polynomial equations with multiple roots:

Polynomial equations with symbolic coefficients:

Transcendental equations, solvable using inverse functions:

Transcendental equations, solvable using special function zeros:

Transcendental inequalities, solvable using special function zeros:

High-degree sparse polynomial equations:

Algebraic equations involving high-degree radicals:

Equations involving non-rational real powers:

Equation with a double root:

Tame elementary function equations:

Elementary function equations in bounded intervals:

Holomorphic function equations in bounded intervals:

Periodic elementary function equations over the reals:

Systems of Real Equations and Inequalities in Several Variables (10)

Quantified polynomial systems:

Transcendental systems, solvable using inverse functions:

Systems exp-log in the first variable and polynomial in the other variables:

Systems elementary and bounded in the first variable and polynomial in the other variables:

Systems analytic and bounded in the first variable and polynomial in the other variables:

Square systems of analytic equations over bounded regions:

Diophantine Equations (11)

Linear systems of equations:

Linear systems of equations and inequalities:

Univariate polynomial equations:

Binary quadratic equations:

Sum of squares equations:

The Pythagorean equation:

Bounded systems of equations and inequalities:

High ‐ degree systems with no solutions:

Transcendental Diophantine systems:

Polynomial systems of congruences:

Modular Equations (4)

Univariate polynomial equations:

Systems of polynomial equations and inequations:

Quantified polynomial systems:

Equations over Finite Fields (3)

Systems of linear equations:

Systems of polynomial equations:

Systems with Mixed Variable Domains (2)

Mixed real and complex variables:

Mixed real and integer variables:

Systems with Geometric Region Constraints (9)

Solve over special regions in 2D:

Solve over special regions in 3D:

A quantified system:

An implicitly defined region:

A parametrically defined region:

Eliminate quantifiers over a Cartesian product of regions:

Regions dependent on parameters:

The answer depends on the parameter value :

Use to specify that is a vector in :

In this case is a vector in :

Generalizations & Extensions (1)

All variables are solved for:

Options (28)

Assumptions (4)

Specify conditions on parameters using Assumptions :

By default, no solutions that require parameters to satisfy equations are produced:

With an equation on parameters given as an assumption, a solution is returned:

Assumptions that contain solve variables are considered to be a part of the system to solve:

Equivalent statement without using Assumptions :

With parameters assumed to belong to a discrete set, solutions involving arbitrary conditions are returned:

Cubics (4)

By default, Solve uses general formulas for solving cubics in radicals only when symbolic parameters are present:

For polynomials with numeric coefficients, Solve does not use the formulas:

With Cubics->False , Solve never uses the formulas:

With Cubics->True , Solve always uses the formulas:

Real roots of irreducible cubics still contain I in their algebraic forms by casus irreducibilis:

Machine-precision numerical evaluation gives a spurious imaginary part:

Arbitrary-precision evaluation still leaves an imaginary part:

With the default setting Cubics  Automatic , explicitly real results are obtained:

GeneratedParameters (1)

Solve may introduce new parameters to represent the solution:

Use GeneratedParameters to control how the parameters are generated:

InverseFunctions (3)

By default, Solve uses inverse functions but prints warning messages:

For symbols with the NumericFunction attribute, symbolic inverses are not used:

With InverseFunctions->True , Solve does not print inverse function warning messages:

Symbolic inverses are used for all symbols:

With InverseFunctions->False , Solve does not use inverse functions:

Solving algebraic equations does not require using inverse functions:

Here, a method based on Reduce is used, as it does not require using inverse functions:

MaxExtraConditions (4)

By default, no solutions requiring extra conditions are produced:

Unless the parameters are discrete:

The default setting, MaxExtraConditions->0 , gives no solutions requiring conditions:

MaxExtraConditions->1 gives solutions requiring up to one equation on parameters:

MaxExtraConditions->2 gives solutions requiring up to two equations on parameters:

Give solutions requiring the minimal number of parameter equations:

Give all solutions:

By default, Solve drops inequation conditions on continuous parameters:

With MaxExtraConditions->All , Solve includes all conditions:

MaxRoots (4)

Find out of roots of a polynomial:

Find out of roots of a polynomial system:

Find solutions of a transcendental system:

When the system contains symbolic parameters, the option value is ignored:

Method (1)

By default, Solve uses inverse functions to solve non-polynomial complex equations:

With Method->Reduce , Solve uses Reduce to find the complete solution set:

Modulus (2)

Solve equations over the integers modulo 9:

Find a modulus for which a system of equations has a solution:

Quartics (3)

By default, Solve uses the general formulas for solving quartics in radicals only when symbolic parameters are present:

For polynomials with numeric coefficients, Solve does not use the formulas:

With Quartics->False , Solve never uses the formulas:

With Quartics->True , Solve always uses the formulas:

VerifySolutions (1)

Solve verifies solutions obtained using non-equivalent transformations:

With VerifySolutions->False , Solve does not verify the solutions:

Some of the solutions returned with VerifySolutions->False are not correct:

This uses a fast numeric test in an attempt to select correct solutions:

In this case numeric verification gives the correct solution set:

WorkingPrecision (1)

By default, Solve finds exact solutions of equations:

Computing the solution using 100-digit numbers is faster:

The result agrees with the exact solution in the first 100 digits:

Computing the solution using machine numbers is much faster:

The result is still quite close to the exact solution:

Applications (7)

Solve a quadratic equation:

Find intersection points of a circle and a parabola:

Find conditions for a quartic to have all roots equal:

A method using quantifier elimination:

Plot a space curve given by an implicit description:

Plot the projection of the space curve on the < x , y >plane:

Find a Pythagorean triple:

Find a sequence of Pythagorean triples:

Find how to pay $2.27 postage with 10-, 23-, and 37-cent stamps:

The same task can be accomplished with IntegerPartitions :

Find 200 roots of a complex analytic function:

Show the roots on the complex plot for the function:

Properties & Relations (15)

Solutions satisfy the equations:

Solutions are given as replacement rules and can be directly used for substitution:

Solve uses < >to represent the empty solution or no solution:

Solve uses < < >> to represent the universal solution or all points satisfying the equations:

For univariate equations, Solve repeats solutions according to their multiplicity:

Solutions of algebraic equations are often given in terms of Root objects:

Use N to compute numeric approximations of Root objects:

Root objects may involve parameters:

Use Series to compute series expansions of Root objects:

The series satisfies the equation up to order 11:

Solve represents solutions in terms of replacement rules:

Reduce represents solutions in terms of Boolean combinations of equations and inequalities:

Solve uses fast heuristics to solve transcendental equations, but may give incomplete solutions:

Reduce uses methods that are often slower, but finds all solutions and gives all necessary conditions:

Use FindInstance to find solution instances:

Like Reduce , FindInstance can be given inequalities and domain specifications:

Use DSolve to solve differential equations:

Use RSolve to solve recurrence equations:

SolveAlways gives the values of parameters for which complex equations are always true:

The same problem can be expressed using ForAll and solved with Solve or Reduce :

Resolve eliminates quantifiers, possibly without solving the resulting quantifier-free system:

Eliminate eliminates variables from systems of complex equations:

This solves the same problem using Resolve :

Reduce and Solve additionally solve the resulting equations:

is bijective iff the equation has exactly one solution for each :

Use FunctionBijective to test whether a function is bijective:

Use FunctionAnalytic to test whether a function is analytic:

An analytic function can have only finitely many zeros in a closed and bounded region:

Possible Issues (9)

Solve gives generic solutions; solutions involving equations on parameters are not given:

Reduce gives all solutions, including those that require equations on parameters:

With MaxExtraConditions->All , Solve also gives non-generic solutions:

Solve results do not depend on whether some of the input equations contain only parameters. The following two systems are equivalent and have no generic solutions:

Use MaxExtraConditions to specify the number of parameter conditions allowed:

Use the Exists quantifier to find solutions that are valid for some value of parameter :

Solve does not eliminate solutions that are neither generically correct nor generically incorrect:

The solutions are correct for and incorrect for :

For transcendental equations, Solve may not give all solutions:

Use Reduce to get all solutions:

Solve with Method->"Reduce" uses Reduce to find solutions, but returns replacement rules:

Using inverse functions allows Solve to find some solutions fast:

Finding the complete solution may take much longer, and the solution may be large:

This finds the values of n for which x ==2 is a solution:

Interpretation of assumptions depends on their syntactic properties. Here the solution is generic in the parameter space restricted by the assumptions:

This mathematically equivalent assumption contains the solve variable, and hence is treated as a part of the system to solve:

There are no generic solutions, because the input is interpreted as:

The solution is non-generic, since it requires the parameters to satisfy an equation:

When parameters are restricted to a discrete set, the notion of genericity is not well defined, and all solutions are returned:

Removable singularities of input equations are generally not considered valid solutions:

However, solutions may include removable singularities that are cancelled by automatic simplification:

The removable singularity at is cancelled by evaluation:

Here the removable singularity at is cancelled by Together , which is used to preprocess the equation:

The value of MaxRoots is used only for systems with numeric coefficients:

When symbolic parameters are present, the option value is ignored:

Expressions given as variables are treated as atomic objects and not as functions of their subexpressions:

Effectively, variables are replaced with new symbols before the equations are solved:

The result comes from:

See Also

Tech Notes

▪
• Symbolic Mathematics: Basic Operations ▪
• Solving Equations ▪
• Simultaneous Equations ▪
• Solving Equations Involving Power Series ▪
• Solving Linear Systems ▪
• Solving Logical Combinations of Equations ▪
• Generic and Non ‐ Generic Solutions ▪
• Eliminating Variables

Related Guides

▪
• Equation Solving ▪
• Polynomial Algebra ▪
• Polynomial Equations ▪
• Solvers over Regions ▪
• Polynomial Systems ▪
• Plane Geometry ▪
• Solid Geometry ▪
• Polygons ▪
• Precollege Education ▪
• Symbolic Vectors, Matrices and Arrays ▪
• Finite Mathematics ▪
• Finite Fields ▪
• Polyhedra ▪
• Geometric Computation ▪
• Additive Number Theory

History

Introduced in 1988 (1.0) | Updated in 1996 (3.0) ▪ 2014 (10.0) ▪ 2020 (12.2) ▪ 2024 (14.0)