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Quadratic Formula Calculator

Solve ax² + bx + c = 0 instantly with step-by-step work.

x = (−b ± √(b² − 4ac)) / 2a

Enter Coefficients

For the equation ax² + bx + c = 0, enter the values of a, b, and c.

Equation: x² − 5x + 6 = 0

About the Quadratic Formula

The quadratic formula solves any equation of the form ax² + bx + c = 0. The discriminant (b² − 4ac) tells you how many real solutions exist before you even solve:

  • Discriminant > 0: Two distinct real roots
  • Discriminant = 0: One real root (a "double root")
  • Discriminant < 0: No real roots (two complex/imaginary roots)

Quadratic Equation FAQ

What is a quadratic equation?
A quadratic equation is any polynomial equation of degree 2 — meaning the highest exponent of x is 2. Standard form is ax² + bx + c = 0 where a ≠ 0. If a = 0, it becomes linear. Quadratics appear in physics, engineering, finance, and many areas of math.
What are the roots of a quadratic?
The roots (also called solutions or zeros) are the x-values where the parabola crosses the x-axis — where y = 0. A quadratic can have 0, 1, or 2 real roots depending on the discriminant.
When should I use the quadratic formula vs. factoring?
Factoring is faster when the equation factors neatly (e.g., x² − 5x + 6 = (x−2)(x−3)). Use the quadratic formula when factoring isn't obvious, when coefficients are large, or when you need exact decimal answers. The formula always works.
What does it mean if there are no real roots?
When the discriminant is negative, the parabola never crosses the x-axis — it floats entirely above or below zero. The solutions are complex numbers involving i (the square root of −1), which are used in advanced math and electrical engineering.