SolveAlways [eqns,vars]
gives the values of parameters that make the equations eqns valid for all values of the variables vars.
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
open all close allBasic 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)
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:
See Also
Tech Notes
Related Guides
History
Introduced in 1988 (1.0)
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: 05-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_solvealways, organization={Wolfram Research}, title={SolveAlways}, year={1988}, url={https://reference.wolfram.com/language/ref/SolveAlways.html}, note=[Accessed: 05-January-2026]}