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

• Operations: Addition, subtraction, multiplication, and division follow algebraic rules, using to simplify.

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 and solve for .

• Quadratic Formula:

• Completing the Square: Rewrite in the form and solve for .

• Discriminant: determines the nature of the roots (real and distinct, real and repeated, 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.

• Area Problems: Setting up equations based on geometric relationships.

• Projectile Motion: Height as a function of time:

• Maxima/Minima: The vertex of the parabola gives the maximum or minimum value.

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 use , , , or instead of .

• Solving Linear Equations: Isolate using inverse operations.

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

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

• Interval Notation: Use for open intervals, for closed intervals.

Example: Solve

Basics of Functions & Their Graphs

Definitions and Properties

A function is a relation that assigns exactly one output to each input. The domain is the set of possible inputs, and the range is the set of possible outputs.

• Function Notation: denotes the output for input .

• 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 Domain and Range: Identify all valid (domain) and (range) values.

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

Challenging Problems

Sample Problems

• Evaluate: (Simplify your answer)

• Rectangle Problem: If the long side is 1 inch longer than the short side and the area is 45 square inches, find the dimensions.

• Solve:

Study Tips

• Start with topics you find most challenging.

• Use available resources: textbooks, online tools, study groups, and office hours.

• Practice regularly and review mistakes to reinforce understanding.

• Work with classmates to discuss and solve different types of problems.

• Simulate test conditions with practice exams to build confidence.

Suggested Exercises