Equations and Solutions

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, solving equations is a fundamental skill, including linear, quadratic, and radical equations.

• Linear Equations: An equation of the form ax + b = 0. The solution is .

• Quadratic Equations: An equation of the form ax^2 + bx + c = 0. Solutions are found using the quadratic formula:

• Radical Equations: Equations involving roots, such as or , often require squaring both sides to eliminate the radical.

Example: Solve and find the sum of the solutions.

• Rewrite as

• Apply the quadratic formula:

• Sum of solutions for is

Functions and Their Properties

Function Notation and Evaluation

A function is a relation that assigns each input exactly one output. Function notation is written as , where is the input variable.

• Evaluating Functions: Substitute the given value into the function.

• Example: If , then .

Domain of a Function

The domain of a function is the set of all possible input values (typically ) for which the function is defined.

• For rational functions , the domain excludes .

• For square root functions , the domain is .

Odd and Even Functions

Odd functions satisfy for all in the domain. Even functions satisfy .

• Example: is odd; is even.

Algebraic Manipulation

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms, factoring, and rationalizing denominators.

• Example: Simplify as .

Factoring

Factoring is expressing an expression as a product of its factors.

• Example:

Systems of Equations

Solving Systems

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Methods: Substitution, elimination, and graphical methods.

• Example: Solve , , .

Graphing and Transformations

Graphing Linear Equations

The graph of a linear equation is a straight line with slope and y-intercept .

• Finding the Equation of a Line: Use the point-slope form .

• Parallel Lines: Have the same slope.

Transformations of Functions

Transformations include shifting, stretching, and compressing graphs.

• Vertical Shifts: shifts up by units.

• Horizontal Shifts: shifts right by units.

• Stretch/Compression: stretches vertically by factor .

Inverse Functions

Finding the Inverse

The inverse function reverses the effect of . To find the inverse, solve for in terms of .

• Example: For , set , solve for :

• So,

Summary Table: Function Transformations

Additional info:

• Some questions involve finding the sum of solutions, which for quadratics is .

• Questions on function composition: means substitute into .

• Domain questions require identifying values that make denominators zero or radicands negative.