Quadratic Calculator

Solve quadratic equations of the form ax² + bx + c = 0. Get exact roots, discriminant, vertex, and full breakdown including complex solutions.

Quadratic Equation Formula

A quadratic equation has the general form:

ax² + bx + c = 0

Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

Discriminant (Δ)

Δ = b² - 4ac
  • Δ > 0 → two real roots
  • Δ = 0 → one real repeated root
  • Δ < 0 → complex roots

Vertex of a Quadratic

h = -b / (2a)
k = f(h)

Common Applications

  • Projectile motion
  • Optimization problems
  • Engineering & physics
  • Curve modeling

Frequently Asked Questions

Yes, when Δ is negative, roots will be imaginary of the form a ± bi.
If a = 0, the equation is not quadratic. It becomes a linear equation bx + c = 0.
The vertex gives the maximum or minimum point of the parabola depending on whether it opens up or down.