Solutions are given as lists of replacements:

UseReplaceAll (/.) 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 fory as a function of the parametera:

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

Replacex by solutions and simplify the results:

UseNormal to remove the conditions:

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

List the first 10 solutions:

Polynomial equations solvable in radicals:

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

By default,Solve usesRoot 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:

Algebraic equations:

Transcendental equations:

Solve cannot find all solutions here:

Find three solutions:

Univariate elementary function equations over bounded regions:

Univariate holomorphic function equations over bounded regions:

HereSolve 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:

Symbolic functions:

Nonanalytic complex equations:

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:

Algebraic equations:

Transcendental equations:

Find a specified number of solutions of transcendental equations:

Square analytic systems over bounded boxes:

Nonanalytic equations:

Polynomial equations:

Polynomial equations with multiple roots:

Polynomial equations with symbolic coefficients:

Algebraic equations:

Piecewise equations:

Transcendental equations, solvable using inverse functions:

Transcendental equations, solvable using special function zeros:

Transcendental inequalities, solvable using special function zeros:

Exp-log equations:

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:

Linear systems:

Polynomial systems:

Quantified polynomial systems:

Algebraic systems:

Piecewise systems:

Transcendental systems, solvable using inverse functions:

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

Quantified system:

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

Quantified system:

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

Quantified system:

Square systems of analytic equations over bounded regions:

Linear systems of equations:

Linear systems of equations and inequalities:

Univariate polynomial equations:

Binary quadratic equations:

Thue 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:

Linear systems:

Univariate polynomial equations:

Systems of polynomial equations and inequations:

Quantified polynomial systems:

Univariate equations:

Systems of linear equations:

Systems of polynomial equations:

Mixed real and complex variables:

Mixed real and integer variables:

Solve over special regions in 2D:

Plot it:

Solve over special regions in 3D:

Plot it:

A quantified system:

An implicitly defined region:

A parametrically defined region:

Derived regions:

Plot it:

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:

Specify conditions on parameters usingAssumptions:

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 usingAssumptions:

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

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:

WithCubics->False,Solve never uses the formulas:

WithCubics->True,Solve always uses the formulas:

Real roots of irreducible cubics still containI in their algebraic forms bycasus irreducibilis:

Machine-precision numerical evaluation gives a spurious imaginary part:

Arbitrary-precision evaluation still leaves an imaginary part:

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

Solve may introduce new parameters to represent the solution:

UseGeneratedParameters to control how the parameters are generated:

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

For symbols with theNumericFunction attribute, symbolic inverses are not used:

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

Symbolic inverses are used for all symbols:

WithInverseFunctions->False,Solve does not use inverse functions:

Solving algebraic equations does not require using inverse functions:

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

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:

WithMaxExtraConditions->All,Solve includes all conditions:

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:

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

WithMethod->Reduce,Solve usesReduce to find the complete solution set:

Solve equations over the integers modulo 9:

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

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:

WithQuartics->False,Solve never uses the formulas:

WithQuartics->True,Solve always uses the formulas:

Solve verifies solutions obtained using non-equivalent transformations:

WithVerifySolutions->False,Solve does not verify the solutions:

Some of the solutions returned withVerifySolutions->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:

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: