Quadratic Equations

A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation is

where a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.

One method for solving a quadratic equation involves rewriting it as a product of two linear expressions. This factored form is written as

In this expression, r and s represent the values that make each factor equal to zero. According to the zero-product property, if the product of two expressions equals zero, then at least one must be zero. As a result, the values represented by r and s are the specific values of the variable that satisfy the equation. Substituting either of these values into the equation produces a true statement, confirming them as solutions.

When factoring is not applicable, a more general solution approach is used by applying the standard formula:

This formula yields all possible equation solutions by substituting the values from the original expression. The term inside the square root, called the discriminant, determines the nature of the solutions. A positive discriminant results in two distinct solutions, a zero discriminant gives one repeated solution, and a negative discriminant indicates that the equation has no real solutions. These methods provide a complete and systematic approach to solving any quadratic equation.