Quadratic Equations

Definition and Fundamental Concepts

A quadratic equation in the variable x is an equation of the form , where a, b, and c are real numbers and a \neq 0. Quadratic equations are also known as second-degree polynomial equations.

• Standard Form:

• Degree: The highest power of x is 2.

Zero-Product Principle

The Zero-Product Principle states that if a product of two or more factors equals zero, then at least one of the factors must be zero.

• Application: Used to solve quadratic equations by factoring.

• Example: Solve by factoring.

• Steps:

  1. Factor the quadratic expression.

  2. Set each factor equal to zero.

  3. Solve for x.

Square Root Property

The Square Root Property is used to solve equations of the form .

• Property: If , then .

• Example: Solve by the square root property.

• Steps:

  1. Isolate the squared term.

  2. Take the square root of both sides.

  3. Include both positive and negative roots.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form .

• Steps:

  1. Move the constant term to the other side.

  2. Divide by the coefficient of if necessary.

  3. Add to both sides to complete the square.

  4. Rewrite as a squared binomial.

  5. Solve using the square root property.

• Example: Solve by completing the square.

The Quadratic Formula

The Quadratic Formula provides a general solution to any quadratic equation .

• Formula:

• Example: Solve using the quadratic formula.

• Steps:

  1. Identify a, b, and c.

  2. Substitute into the formula.

  3. Calculate the discriminant .

  4. Find the solutions.

The Discriminant

The discriminant of a quadratic equation is . It determines the number and type of solutions.

• If : Two distinct real solutions.

• If : One real solution (a repeated root).

• If : Two complex (non-real) solutions.

• Example: Compute the discriminant for and determine the number and type of solutions.

Applications: Modeling with Quadratic Equations

Quadratic equations can be used to model real-world phenomena, such as blood pressure as a function of age.

• Example: The formula models a woman’s normal systolic blood pressure, P, at age A.

• Application: To find the age for a given blood pressure, set to the desired value and solve for using quadratic methods.

• Example: Find the age, to the nearest year, of a woman whose normal systolic blood pressure is 115 mm Hg.

Summary Table: Methods for Solving Quadratic Equations

Additional info: Examples and applications were expanded for clarity and completeness. The notes cover all standard methods for solving quadratic equations, including their use in modeling real-world scenarios.