Quadratic Equation Solver (ax² + bx + c = 0)
Solve second-degree quadratic equations instantly with step-by-step mathematical proofs. Computes real and complex roots, discriminant evaluation, parabolic vertex coordinates, and axis of symmetry.
Step-by-Step Derivation:
The Quadratic Formula: Derivation, Discriminant & Geometry
A quadratic equation is any polynomial equation of the second degree, meaning the highest exponent of the variable x is 2. The standard form is expressed as ax² + bx + c = 0, where a, b, and c are constant coefficients, and a ≠ 0.
The Universal Quadratic Formula
Derived by completing the square on the general quadratic equation, the formula provides exact solutions for any polynomial:
The Role of the Discriminant (Δ)
The expression beneath the radical sign, Δ = b² - 4ac, is termed the discriminant. It fundamentally determines the nature and quantity of the roots without having to complete the full calculation:
- Δ > 0: The equation has two distinct real roots. Graphically, the parabola intersects the x-axis at two distinct points.
- Δ = 0: The equation has one repeated real root (x = -b/2a). The parabola touches the x-axis tangentially at its vertex.
- Δ < 0: The equation has two complex conjugate roots involving the imaginary unit i (√-1). The parabola never crosses the real x-axis.
Vertex and Projectile Motion in Physics
In physics, quadratic functions model the trajectory of ballistic projectiles under Earth's gravitational acceleration: y(t) = -½gt² + v₀t + y₀. The vertex of the parabola, given by h = -b / (2a), determines the exact time at which a projectile reaches its maximum peak altitude.