Solving Equations (Review and Preview)

The domain convention

The domain of a variable in a given expression is the set of all real-number values of the variable for which the expression is defined.

Terminology

· Equations that become true statements for all values in the domain of the variable are called identities.

· A conditional equation is true only for some (or perhaps none) of the values of the variable.

· Two equations are equivalent when they have exactly the same solutions.

Extraneous Solutions

Multiplying both sides of an equation by an expression involving the variable may introduce extraneous solutions that do not check in the original equation. Therefore, it is always necessary to check any candidates for solutions that you obtain in this manner.

Zero-Product Property of Real Numbers

pq = 0 if and only if p = 0 or q = 0 (or both)

The Quadratic Formula

The solutions if the quadratic equation, where, are given by