Quadratic Equation Solver (Detailed) – Tutorial

Quadratic Equation (ax² + bx + c = 0) — Detailed Guide

Master quadratic equations: finding roots, the discriminant (Δ), parabola geometry, vertex, axis of symmetry, and factoring.

1) What is a quadratic equation?

General form: ax² + bx + c = 0 (a ≠ 0)

If a = 0 it reduces to a linear equation. Geometrically, f(x) = ax² + bx + c is a parabola; roots are the x-coordinates where it crosses the x-axis.

2) Discriminant (Δ)

Δ = b² − 4ac

• Δ > 0 → 2 distinct real roots
• Δ = 0 → one double root
• Δ < 0 → no real roots (complex roots)

3) Root formula

x = (−b ± √Δ) / (2a)

4) Vertex and parabola

• x₀ = −b / (2a)
• y₀ = f(x₀)
• a > 0 → parabola opens upward (minimum)
• a < 0 → parabola opens downward (maximum)

5) Vieta's Formulas

• x₁ + x₂ = −b/a
• x₁ · x₂ = c/a

Note: For educational purposes only.