Equations and Inequalities

Quadratic Equations

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

where a, b, and c are real numbers, with a ≠ 0. Quadratic equations are also called second-degree polynomial equations in x.

Factoring and the Zero Product Principle

Factoring a polynomial means expressing it as a product of simpler polynomials. Common methods include factoring out the greatest common factor (GCF) and factoring trinomials.

• Zero Product Principle: If the product of two algebraic expressions is zero, then at least one of the factors must be zero.

Steps to Solve a Quadratic Equation by Factoring:

1. Rewrite the equation in the form .

2. Factor the quadratic expression completely.

3. Set each factor containing a variable equal to zero.

4. Solve the resulting linear equations.

5. Check the solutions in the original equation.

Example: Solve by factoring. Factor: Set each factor to zero: or Solutions: or

Solving Quadratic Equations by the Square Root Property

The square root property states that if , then .

• Isolate the squared term.

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

Example: Solve

Solving Quadratic Equations by Completing the Square

Completing the square is a method used to solve quadratic equations by converting the equation into a perfect square trinomial.

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

• Add the square of half the coefficient of to both sides to create a perfect square trinomial.

• Factor the perfect square trinomial.

• Solve for using the square root property.

Example: Solve by completing the square. Move constant: Add to both sides: Factor:

The Quadratic Formula

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

This formula is derived by completing the square on the general quadratic equation.

Choosing the Most Efficient Method

When solving quadratic equations, choose the method based on the equation's structure:

• Factoring: Use when the equation factors easily.

• Square Root Property: Use when the equation is in the form .

• Completing the Square: Use when the coefficient of is 1 and the equation does not factor easily.

• Quadratic Formula: Use for any quadratic equation, especially when factoring is difficult or impossible.

Applications of Quadratic Equations

Quadratic equations are used to model a variety of real-world problems, such as projectile motion, area, and optimization problems.

Example: A man’s normal systolic blood pressure, , at age , is modeled by the equation . Find the age when mm Hg.

Set up the equation: Solve using the quadratic formula.

Geometric Applications: The Pythagorean Theorem

Quadratic equations often arise in geometry, especially with right triangles. The Pythagorean Theorem states:

where and are the legs and is the hypotenuse of a right triangle.

Example: Wheelchair Ramp Problem

A wheelchair ramp with a length of 122 inches has a horizontal distance of 120 inches. What is the ramp’s vertical distance?

Let be the vertical distance. By the Pythagorean Theorem:

Solve for :

Application: Construction laws require every vertical rise of 1 inch to have a horizontal run of 12 inches. Check if the ramp meets this requirement by comparing the calculated vertical distance to the horizontal run.

Example: Right Triangle Property Problem

A piece of property has the shape of a right triangle. The longer leg is 20 m longer than twice the length of the shorter leg. The hypotenuse is 10 m longer than the length of the longer leg. Find the lengths of the sides.

• Let the shorter leg be .

• Longer leg:

• Hypotenuse:

Apply the Pythagorean Theorem:

Expand and solve the resulting quadratic equation for .

Summary Table: Methods for Solving Quadratic Equations