Quadratic Equations

Definition and Identification

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

where a, b, and c are real numbers, and a \neq 0. Quadratic equations are fundamental in algebra and appear in many mathematical and real-world contexts.

• Example 1: (Quadratic equation)

• Example 2: (Quadratic equation; can be rewritten as )

• Example 3: (Not quadratic; degree is 1)

• Example 4: (Quadratic equation; can be rewritten as )

Methods for Solving Quadratic Equations

Solving by Factoring

Factoring is a method where the quadratic expression is rewritten as a product of two binomials. This method is effective when the equation can be easily factored.

• Step 1: Write the equation in standard form ().

• Step 2: Factor the quadratic expression.

• Step 3: Set each factor equal to zero and solve for .

Example: Solve

• Factor:

• Set each factor to zero: or

• Solutions: ,

Solving by Square Root Property

This method is used when the equation can be written so that (or ) is isolated.

• Step 1: Isolate the term.

• Step 2: Take the square root of both sides, remembering to include both positive and negative roots.

Example: Solve

• Take square root:

Example: Solve

• Take square root:

• Solutions: ,

Solving by Completing the Square

Completing the square is a method that rewrites the quadratic equation so that one side is a perfect square trinomial.

• Step 1: Write the equation in standard form and isolate and terms.

• Step 2: Add a value to both sides to complete the square.

• Step 3: Rewrite as a squared binomial.

• Step 4: Solve using the square root property.

Example: Solve

• Move constant:

• Left side is , right side is $4$

• Take square root:

• Solutions: ,

Solving by Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation.

• Step 1: Write the equation in standard form ().

• Step 2: Identify , , and .

• Step 3: Use the quadratic formula:

Example: Solve

• Rewrite:

• , ,

• Plug into formula:

Since the discriminant is negative, solutions are complex.

The Discriminant

Definition and Properties

The discriminant is the part of the quadratic formula under the square root: . It determines the number and type of solutions for a quadratic equation.

• Two distinct real solutions:

• One real solution (repeated root):

• Two complex (non-real) solutions:

Example: For , (two complex solutions).