Calculator That Solves Quadratic Equations

Enter the coefficients (a, b, and c) for the quadratic equation ax² + bx + c = 0.

Understanding Quadratic Equations and Their Solutions

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).
• a is the coefficient of the squared term (x²), and it cannot be zero (a ≠ 0). If a = 0, the equation becomes a linear equation.
• b is the coefficient of the linear term (x).
• c is the constant term (also called the y-intercept when graphing).

The Quadratic Formula

The solutions to a quadratic equation, also known as its roots, are the values of x that satisfy the equation. These roots can be found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

The Discriminant (Δ)

A crucial part of the quadratic formula is the expression under the square root sign, known as the discriminant (Δ). It is calculated as:

Δ = b² - 4ac

The value of the discriminant tells us about the nature of the roots:

• If Δ > 0: The equation has two distinct real roots.
• If Δ = 0: The equation has exactly one real root (a repeated or double root).
• If Δ < 0: The equation has two complex conjugate roots (involving imaginary numbers).

How the Calculator Works

This calculator takes the coefficients a, b, and c you provide and plugs them into the quadratic formula. It first calculates the discriminant to determine the nature of the roots. Then, based on the discriminant's value, it computes and displays the real or complex roots. If a is zero, it indicates that the equation is not quadratic.

Use Cases

Quadratic equations appear in various fields:

• Physics: Describing projectile motion (e.g., the path of a ball thrown into the air).
• Engineering: Calculating areas, optimizing designs, and analyzing circuits.
• Economics: Modeling profit and cost functions.
• Geometry: Finding dimensions of shapes or solving problems involving areas and lengths.
• General Problem Solving: Many real-world problems can be translated into quadratic equations.