Complex Numbers and Operations

Standard Form of Complex Numbers

Complex numbers are expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, defined by .

• Definition: A complex number is any number of the form .

• Example: is a complex number.

• Standard Form: Always write complex numbers as .

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules and the property .

• Addition/Subtraction: Combine like terms: .

• Multiplication: Use distributive property and .

• Conjugate: The conjugate of is .

• Example:

• Product with Conjugate: and its conjugate .

Division and Standard Form

To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.

• Formula:

• Example:

Square Roots of Negative Numbers

Square roots of negative numbers are expressed using the imaginary unit .

• Formula: for

• Example:

Quadratic Equations and Properties

Solving Quadratic Equations

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

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

• Square Root Property: If , then .

• Completing the Square: Transform into .

• Quadratic Formula:

• Example:

Perfect Square Trinomials

A perfect square trinomial is of the form .

• To complete the square: Add to .

• Example: ; add .

Applications of Quadratic Equations

Projectile Motion

Quadratic equations model the height of an object in vertical motion.

• Formula:

• Example: models a ball thrown upward.

Area Problems

Quadratic equations can be used to solve geometric problems involving area.

• Example: If the length of a garden is 7 feet longer than its width and the area is 78 square feet, set up .

Polynomial Equations

Solving Higher-Degree Equations

Equations of degree higher than two can be solved by factoring, substitution, or other algebraic methods.

• Example:

• Substitution: For equations like , let .

Radical and Rational Equations

Solving Radical Equations

Radical equations contain variables inside a root. Isolate the radical and square both sides to solve.

• Example:

Solving Rational Inequalities

Rational inequalities involve fractions with polynomials in the numerator and denominator. Solutions are often expressed in interval notation.

• Steps:

  1. Find critical points by setting numerator and denominator to zero.

  2. Test intervals between critical points.

  3. Express solution in interval notation.

• Example:

Summary Table: Methods for Solving Equations

Additional info:

• Some problems involve application of algebraic concepts to real-world scenarios, such as projectile motion and area calculations.

• Substitution is a useful technique for solving higher-degree polynomials.

• Interval notation is used to express solutions to inequalities.