Quadratic Equation Calculator

A quadratic equation is a polynomial equation of the second degree, where the highest power of the variable is 2. Standard Form: ax² + bx + c = 0 Where: • a, b, c = constants (coefficients) • x = variable (unknown) • a ≠ 0 (if a = 0, equation becomes linear) Three Essential Conditions: 1. Polynomial Equation: No variables in denominators or under radical signs 2. One Unknown Variable: Only one letter represents the unknown 3. Highest Degree is 2: After simplification, highest exponent is 2, and a ≠ 0 Components: • ax²: Quadratic term (a is the leading coefficient, must be non-zero) • bx: Linear term (b can be 0, creating ax² + c = 0) • c: Constant term (c can be 0, creating ax² + bx = 0) Solution Methods: 1. Quadratic Formula (Universal Method): x = [-b ± √(b² - 4ac)] / 2a 2. Factoring: If ax² + bx + c = (px + q)(rx + s), then x = -q/p or x = -s/r 3. Completing the Square: Convert to (x + p)² = q form 4. Graphing: Find x-intercepts of parabola y = ax² + bx + c The Discriminant: Δ = b² - 4ac Determines nature of roots: • Δ > 0: Two distinct real roots • Δ = 0: One repeated real root (double root) • Δ < 0: Two complex conjugate roots (no real solutions) Types of Solutions: 1. Two Real Roots: x₁ and x₂ (when Δ > 0) 2. One Real Root: x (when Δ = 0, also called repeated/double root) 3. Two Complex Roots: x = p ± qi (when Δ < 0) Special Forms: • Pure Quadratic: ax² + c = 0 (b = 0) • Incomplete Quadratic: ax² + bx = 0 (c = 0) • Monic Quadratic: x² + bx + c = 0 (a = 1) Key Properties: • Sum of roots: x₁ + x₂ = -b/a • Product of roots: x₁ × x₂ = c/a • Vertex form: a(x - h)² + k, where (h, k) is the vertex • Axis of symmetry: x = -b/(2a) Applications: Projectile motion, area optimization, profit maximization, physics (kinematic equations), engineering, architecture (parabolic curves), and any scenario involving squared relationships.