Formula (Algebra)
An algebraic formula is a rule expressed as an equation that shows how variables are related, allowing you to calculate one quantity when others are known.
Definition
A formula is a math rule written with letters and symbols that shows how to calculate something: you plug in the values you know to find the value you want, saving you from having to re-derive the same rule every time. A formula is an equation expressing a relationship among variables that can be used to compute one variable given the others, and it is really a type of literal equation, so mastering literal equations means you can always rearrange a formula rather than memorize a new one for each variable you need to solve for. In the algebraic sense, a formula is an equation in multiple variables that encodes a functional relationship; formulas are special cases of identities (true for all values in a domain) or equations defining implicit functions, and in mathematical logic, a formula is a well-formed expression in a formal language that evaluates to true or false.
Example
Area of a rectangle: $A = l \times w$; if the length is $8$ and the width is $5$, then $A = 8 \times 5 = 40$ square units. Distance formula: $d = rt$; if $r = 60$ mph and $t = 2.5$ hours, then $d = 150$ miles, and rearranging for time gives $t = d/r$. The quadratic formula $x = (-b \pm \sqrt{b^2-4ac})/(2a)$ is derived by completing the square on $ax^2 + bx + c = 0$ and is valid for all $a \neq 0$ over $\mathbb{R}$ (or $\mathbb{C}$).
Key Insight
Formulas save you from having to re-derive the same rule every time, and every science formula is a literal equation that can be rearranged to solve for any variable. Formulas encode compressed knowledge: the quadratic formula contains the complete theory of solutions for degree-$2$ polynomials, since the discriminant $b^2 - 4ac$ determines the number and type of roots without even computing them.