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SolveAlways[eqns,vars]

gives the values of parameters that make the equations eqns valid for all values of the variables vars.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
WorkingPrecision  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
History
Cite this Page

SolveAlways

SolveAlways[eqns,vars]

gives the values of parameters that make the equations eqns valid for all values of the variables vars.

Details and Options

  • Equations are given in the form lhs==rhs.
  • Simultaneous equations can be combined either in a list or with &&.
  • A single variable or a list of variables can be specified.
  • SolveAlways works primarily with linear and polynomial equations.
  • SolveAlways produces relations between parameters that appear in eqns, but are not in the list of variables vars.
  • SolveAlways[eqns,vars] is equivalent to Solve[!Eliminate[!eqns,vars]].

Examples

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Basic Examples  (1)

Find a condition for the equation to hold for any value of x:

Scope  (6)

A univariate polynomial equation:

A multivariate polynomial equation:

A list of polynomial equations:

An equation involving radicals:

An inequation:

Boolean combinations of equations and inequations:

Options  (1)

WorkingPrecision  (1)

By default, SolveAlways finds exact solutions:

This computes the solutions using 20-digit numbers:

Applications  (2)

Find a condition for a cubic polynomial to have a triple root:

This solves the same problem using Reduce:

The same problem can also be solved using Subresultants and Solve:

Solve for undetermined coefficients in the series expansion:

Properties & Relations  (1)

Numeric solutions make the equations identically true:

An equivalent formulation using Solve and Eliminate:

An equivalent formulation using Solve and Resolve:

This solves the same problem using Reduce:

Possible Issues  (1)

SolveAlways gives generic solutions:

The equations may not be identically true when parameters satisfy additional equations:

This finds conditions on parameters for which the solutions are not correct:

Wolfram Research (1988), SolveAlways, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveAlways.html.

Text

Wolfram Research (1988), SolveAlways, Wolfram Language function, https://reference.wolfram.com/language/ref/SolveAlways.html.

CMS

Wolfram Language. 1988. "SolveAlways." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SolveAlways.html.

APA

Wolfram Language. (1988). SolveAlways. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SolveAlways.html

BibTeX

@misc{reference.wolfram_2025_solvealways, author="Wolfram Research", title="{SolveAlways}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/SolveAlways.html}", note=[Accessed: 28-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_solvealways, organization={Wolfram Research}, title={SolveAlways}, year={1988}, url={https://reference.wolfram.com/language/ref/SolveAlways.html}, note=[Accessed: 28-August-2025]}