Section 1.4: Quadratic Equations

This section covers the fundamental methods for solving quadratic equations, including factoring, the square root property, completing the square, and the quadratic formula. It also discusses the discriminant and its role in determining the nature of solutions.

Review: Essential Skills for Quadratic Equations

• Simplifying Radical Expressions Using the Product Rule for Radicals: The product rule states that for non-negative real numbers a and b.

• Rationalizing Denominators: To eliminate radicals from denominators, multiply numerator and denominator by a suitable radical to make the denominator rational.

• Factoring Trinomials: Recognize and factor trinomials with leading coefficients equal to or not equal to 1.

• Simplifying Radicals with Negative Radicands: Use the imaginary unit where , so for .

Quadratic Equations

Definition

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

, where

Objective 1: Solving Quadratic Equations by Factoring and the Zero Product Property

Zero Product Property

• If , then either or (or both).

• To solve a quadratic by factoring, rewrite the equation in standard form, factor, and set each factor equal to zero.

Example:

Solve

• First, rewrite in standard form:

• Factor:

• Set each factor to zero: or

• Solutions: or

Objective 2: Solving Quadratic Equations Using the Square Root Property

Square Root Property

• If , then

• Use this method when the equation can be written as

Examples:

• a.

• b.

• c.

Objective 3: Solving Quadratic Equations by Completing the Square

Perfect Square Trinomials

• Form:

• Form:

Examples:

Relationship: The constant term is always the square of half the coefficient of the linear term.

Making a Perfect Square Trinomial:

• a. Half of is , square it: $36 to complete the square:

• b. Half of $5, square it: . Add $6.25$ to complete the square:

• c. Half of is , square it: . Add $\frac{9}{16}$ to complete the square:

Steps for Completing the Square

1. Move the constant term to the other side of the equation.

2. If , divide both sides by to make the coefficient of equal to 1.

3. Add the square of half the coefficient of to both sides.

4. Write the left side as a squared binomial.

5. Solve for using the square root property.

Examples:

• a. Add $9, ):

• b. Add to both sides:

Objective 4: Solving Quadratic Equations Using the Quadratic Formula

Derivation (Outline)

1. Start with

2. Divide both sides by

3. Move the constant to the right side

4. Complete the square on the left

5. Take the square root of both sides and solve for

Quadratic Formula

The solution to is:

Examples:

• Solve , ,

• Solve , ,

Objective 5: Using the Discriminant to Determine the Type of Solutions

Definition of the Discriminant

The discriminant of a quadratic equation is .

Nature of Solutions Based on the Discriminant

Examples:

• a. Conclusion: Two complex solutions

• b. Rewrite: Conclusion: One real solution

Additional info: These methods are foundational for solving quadratic equations in algebra and are widely applicable in mathematics, science, and engineering.