Calculator with Equation

Quadratic Equation Solver

Understanding the Quadratic Equation and Its Roots

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients (constants), and 'a' cannot be zero (otherwise, it wouldn't be a quadratic equation). The primary goal when solving a quadratic equation is to find the value(s) of 'x' that satisfy the equation, known as the roots or solutions.

The Discriminant

The nature of the roots (real, complex, distinct, or repeated) depends on a value called the discriminant, often denoted by the Greek letter Delta (Δ). It is calculated as:

Δ = b² - 4ac

• If Δ > 0: The equation has two distinct real roots.
• If Δ = 0: The equation has exactly one real root (a repeated root).
• If Δ < 0: The equation has two distinct complex roots (involving the imaginary unit 'i').

The Quadratic Formula

To find the actual roots, we use the quadratic formula. This formula provides the solutions for 'x' based on the coefficients 'a', 'b', and 'c':

x = [-b ± √(b² - 4ac)] / 2a

This can be broken down into two potential solutions:

1. x₁ = [-b + √(Δ)] / 2a
2. x₂ = [-b - √(Δ)] / 2a

Our calculator uses this formula to determine the roots of any quadratic equation you input.

Use Cases

Quadratic equations and their solutions are fundamental in many areas of mathematics, science, and engineering:

• Physics: Modeling projectile motion (e.g., the trajectory of a ball), determining time of flight, and analyzing acceleration.
• Engineering: Designing structures, analyzing electrical circuits, and optimizing processes.
• Economics: Calculating profit maximization, cost analysis, and modeling market behavior.
• Geometry: Finding dimensions of shapes when relationships are described by quadratic functions.
• General Problem Solving: Anywhere a relationship between two variables can be expressed as a quadratic function.