Quadratic Equation Solver

The discriminant (b² - 4ac) determines the nature of the roots. If positive: two distinct real roots. If zero: one repeated real root. If negative: two complex (imaginary) roots — no real solutions. The discriminant tells you what to expect before calculating the full solution.

Factoring works when the equation factors into nice integers — fast but not always possible. Completing the square always works and reveals the vertex form. The quadratic formula always works for any equation. Graphically, the roots are the x-intercepts of the parabola y = ax² + bx + c.

Projectile motion follows a quadratic equation: h = -½gt² + v₀t + h₀, where h is height, g is gravitational acceleration, v₀ is initial velocity, and h₀ is initial height. Solving for t when h = 0 tells you when the projectile hits the ground.