# 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 allclose all

## 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_2024_solvealways, author="Wolfram Research", title="{SolveAlways}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/SolveAlways.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_solvealways, organization={Wolfram Research}, title={SolveAlways}, year={1988}, url={https://reference.wolfram.com/language/ref/SolveAlways.html}, note=[Accessed: 30-May-2024 ]}