Quadratic Equation Solver

Solve quadratic equations ax² + bx + c = 0 with detailed steps

x²

= 0

Solutions

Solution Steps

What is a Quadratic Equation and How to Solve It?

A quadratic equation is a second-degree polynomial equation in a single variable, written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

The Quadratic Formula

To find the roots (the values of x that satisfy the equation), we use the universal quadratic formula:

x = (-b ± √Δ) / 2a

Where the discriminant (Δ) is given by:

Δ = b² - 4ac

Understanding the Discriminant (Δ)

Δ > 0: The equation has two distinct real roots. Geometrically, the parabola crosses the x-axis at two points.
Δ = 0: The equation has exactly one real root (a repeated root). The parabola touches the x-axis at its vertex.
Δ < 0: There are no real roots. The solutions are complex numbers, and the parabola does not touch the x-axis.

Real-World Applications

Quadratic equations are essential in physics for projectile motion, in economics for identifying profit maximization points, and in engineering for designing parabolic arches and mirrors.

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