Complex Numbers

Definition and Properties

Complex numbers extend the real number system by including the imaginary unit i, where . They are written in the form , where a and b are real numbers.

• Imaginary Unit:

• Pure Imaginary Numbers: Numbers of the form where

• Complex Conjugate: For , the conjugate is

• Multiplication and Division: Use the distributive property and rationalize denominators as needed.

Example:

Quadratic Equations

Solving Quadratics

Quadratic equations are equations of the form . They can be solved by factoring, completing the square, or using the quadratic formula.

• Factoring: Express as a product of binomials and set each factor to zero.

• Quadratic Formula:

• Zero Factor Property: If , then or

• Discriminant: determines the nature of the roots (real and distinct, real and equal, or complex).

Example: Solve by factoring:

Common Applications of Quadratic Equations

Word Problems and Modeling

Quadratic equations are used to model various real-world scenarios, such as projectile motion, area problems, and optimization.

• Setting up Equations: Translate word problems into quadratic equations.

• Solving for Unknowns: Use appropriate methods to find solutions relevant to the context.

Example: The area of a rectangle is 45 square inches. If the long side is 1 inch longer than the short side, set up and solve .

Linear Equations and Inequalities

Solving and Graphing

Linear equations are of the form . Linear inequalities involve expressions like or .

• Solving Linear Equations: Isolate the variable using inverse operations.

• Solving Linear Inequalities: Similar to equations, but reverse the inequality when multiplying or dividing by a negative number.

• Graphing Solutions: Represent solutions on a number line or coordinate plane.

• Intersection of Intervals: Find where two or more inequalities are simultaneously true.

Example: Solve

Basics of Functions & Their Graphs

Definitions and Properties

A function is a relation that assigns exactly one output to each input. Key features include domain, range, and the vertical line test.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

• Evaluating Functions: Substitute the input value into the function rule.

• Finding Information from Graphs: Use the graph to determine function values, intercepts, and intervals of increase/decrease.

Example: For , the domain is all real numbers, and the range is .

Challenging Problems

Sample Problems

• Evaluate and simplify your answer.

• Given a rectangle with area 45 square inches and the long side 1 inch longer than the short side, find the dimensions.

• Solve .

Summary Table: Key Concepts

Study Tips

• Start with topics you find most challenging.

• Practice regularly and review mistakes.

• Work with classmates and use available resources.

• Attempt challenging problems for deeper understanding.