= Expression
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= Mathematical expression
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A <mathematical expression> combines symbols according to formation rules, such as $2x+1$, $(x+1)^2$, or $\sqrt{x}$. With values for its <variables> and meanings for its <algebraic operations>, it can be evaluated. An <equation> relates two <mathematical expressions>; an expression alone need not assert anything.
For example, $2x+3$ evaluates to seven at $x=2$ and nine at $x=3$. Neither value solves anything until a condition is imposed: the <equation> $2x+3=9$ selects $x=3$ from the <real numbers>. This distinction is useful when moving from a formula to a question about which inputs satisfy it. Evaluation by replacing the <variables> with assigned values is described in https://openstax.org/books/college-algebra-2e/pages/1-1-real-numbers-algebra-essentials[College Algebra 2e, §1.1: Evaluating Algebraic Expressions].
Different written <mathematical expressions> can have the same value for every input in a common <domain of a function>. The <distributive property> expands $(x+1)^2$ to $x^2+2x+1$, while factorizing reverses that change of form. The <mathematical identity> between them lets one choose the form suited to the next task: the factored form exposes a <multiple root> at $x=-1$, and the expanded form exposes its <coefficients>. Simplifying should preserve values on the stated domain, rather than silently enlarging it.
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