Quadratic Equations and Their Solutions

Definition of a Quadratic Equation

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.

Methods for Solving Quadratic Equations

• Factoring

• Square Root Property

• Quadratic Formula

• Using the Discriminant

Solving Quadratic Equations by Factoring

The Zero-Product Principle

The zero-product principle states that if the product of two algebraic expressions is zero, then at least one of the factors must be zero:

If , then or .

Steps for Solving by Factoring

1. Rewrite the equation in general 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: Solving by Factoring

Suppose we have .

• Step 1: Already in general form.

• Step 2: Factor: .

• Step 3: Set each factor to zero: or .

• Step 4: Solutions: or .

• Step 5: Check in the original equation.

Solving Quadratic Equations by the Square Root Property

Quadratic equations of the form (where u is an algebraic expression and d is a nonzero real number) can be solved using the Square Root Property:

If , then or .

Equivalently, .

Example: Solving by the Square Root Property

Solve :

• Take the square root of both sides: .

• So, or .

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula provides the solutions to any quadratic equation (with ):

Example: Using the Quadratic Formula

Solve :

• Here, , , .

• Plug into the formula:

The Discriminant

The discriminant of a quadratic equation is the expression under the square root in the quadratic formula:

The discriminant determines the number and type of solutions:

• If : Two unequal real solutions

• If : One real (repeated) solution

• If : Two imaginary (complex) solutions

Example: Using the Discriminant

For , , , :

(positive, so two real solutions)

Applications of Quadratic Equations

Modeling with Quadratic Equations

Quadratic equations are used to model various real-world phenomena, such as projectile motion, area problems, and physical relationships.

Example: Blood Pressure Model

The formula models a woman’s normal systolic blood pressure, P, at age A. To find the age for a given blood pressure, solve the quadratic equation for A.

The Pythagorean Theorem

Statement and Formula

The Pythagorean Theorem relates the lengths of the sides of a right triangle. If the legs have lengths a and b, and the hypotenuse has length c, then:

Example: Application to a Wheelchair Ramp

A wheelchair ramp with a length of 122 inches (hypotenuse) and a horizontal distance of 120 inches (one leg) is given. To find the vertical distance (other leg), use the Pythagorean Theorem:

• Let x be the vertical distance.

• Set up the equation:

• Solve for x:

inches

Checking Construction Requirements

Construction laws may require that every vertical rise of 1 inch corresponds to a horizontal run of 12 inches. For a vertical rise of 22 inches, the required horizontal run would be inches. Since the ramp's horizontal run is only 120 inches, it does not satisfy the requirement.

Additional info: The Pythagorean Theorem is a fundamental tool in geometry and algebra for relating the sides of right triangles, and is frequently used in practical applications such as construction, navigation, and physics.