Understanding Quadratic Equations | A Comprehensive Guide to the Basics and Applications

quadratic equation

A quadratic equation is a second-degree polynomial equation in one variable, typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable

A quadratic equation is a second-degree polynomial equation in one variable, typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. The highest power of the variable x is 2, which makes it a quadratic equation.

The general form of a quadratic equation, as mentioned above, consists of three terms: the quadratic term (ax^2), the linear term (bx), and the constant term (c). The term “quadratic” comes from the Latin word quadratus, meaning “square,” as the quadratic term involves squaring the variable.

The solutions or roots of a quadratic equation are the values of x that satisfy the equation. Depending on the discriminant (b^2 – 4ac) of the quadratic equation, there can be three possible scenarios:

1. If the discriminant is positive (greater than zero), then the equation has two distinct real roots. These roots can be found using the quadratic formula: x = (-b ± √(b^2 – 4ac)) / (2a).

2. If the discriminant is zero, then the equation has one real root (also known as a repeated or double root). Again, the quadratic formula can be used to find this root.

3. If the discriminant is negative, then the equation has no real roots. In this case, the roots are complex (non-real) conjugates of the form x = (-b ± i√(4ac – b^2)) / (2a), where i represents the imaginary unit.

Quadratic equations have several applications in mathematics and other fields, such as physics, engineering, and finance. They can be used to solve problems involving motion, such as projectile motion, or to find the optimal solution for certain optimization problems.

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