Quadratic Equations

Definition and Standard Form

A quadratic equation is an equation that can be written in the form:

where a, b, and c are real numbers and . This is also called the standard form of a quadratic equation.

Solving Quadratic Equations by Factoring

Zero-Factor Property

The Zero-Factor Property states:

• If and are complex numbers and , then or (or both).

Square Root Property

The Square Root Property states:

• If , then .

Examples: Solving by Factoring

• Example a: Factor: Solutions: or

• Example b: Rearranged: Factor: Solutions: or

• Example c: Use Square Root Property:

• Example d: Use Square Root Property: Solutions: or

Completing the Square

Method and Examples

Completing the square is a technique used to rewrite a quadratic expression in the form to make it easier to solve.

General form for completing the square:

Practice: Complete the Table

Solving Using the Quadratic Formula

The Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation of the form :

Note: The fraction bar in the quadratic formula extends under the entire numerator, including the and the terms.

Example: Solving by the Quadratic Formula

• Example: Here, , , Substitute into the formula:

Summary Table: Methods for Solving Quadratic Equations

Key Terms: quadratic equation, standard form, zero-factor property, square root property, completing the square, quadratic formula.