Quadratic Equations

Standard Form and Key Properties

Quadratic equations are polynomial equations of degree two, typically written in the standard form:

• Standard Form:

• Square Root Property: If , then

Quadratic equations can be solved using several methods, including factoring, completing the square, and the quadratic formula.

Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation:

• Formula:

• Discriminant: The expression under the square root, , determines the nature of the solutions (real and distinct, real and equal, or complex).

Methods for Solving Quadratic Equations

Factoring

Factoring is used when the quadratic can be written as a product of two binomials:

• Set the equation to zero:

• Factor into

• Set each factor equal to zero and solve for

Example: Solution set:

Completing the Square

Completing the square transforms the quadratic into a perfect square trinomial:

• Move the constant to the other side:

• Add to both sides

• Rewrite as

• Take the square root of both sides and solve for

Example: Add $4x^2 - 4x + 4 = 12(x - 2)^2 = 12x - 2 = \\pm \\sqrt{12}x = 2 \\pm 2\\\sqrt{3}$

Using the Quadratic Formula

Apply the quadratic formula when factoring is not straightforward:

• Identify , , and from

• Substitute into the formula

• Simplify to find the solution set

Example:

Special Cases and Applications

Zero-Factor Property

If a quadratic equation can be factored such that one factor equals zero, the solutions are the roots of each factor.

Example: Solution set:

Square Root Property

Used when the equation is in the form :

• Take the square root of both sides:

Example:

Summary Table: Methods for Solving Quadratic Equations

Additional info:

• Examples in the notes cover all major solution methods for quadratic equations, including cases with irrational and complex solutions.

• Each method is illustrated with step-by-step solutions, reinforcing procedural understanding.