Quadratic Equation Calculator

Solve ax²+bx+c=0 with the discriminant and quadratic formula, including repeated, irrational and complex roots, vertex and axis details.

Solve quadratic equations from coefficients a, b and c

Enter the coefficients from ax² + bx + c = 0 to calculate the discriminant and both roots.

The result also reports whether the roots are real or complex, plus the parabola's axis of symmetry and vertex.

Features

  • Separate inputs for coefficients a, b and c.
  • Discriminant calculation.
  • Two real roots, a repeated root or complex conjugate roots.
  • Quadratic-formula steps.
  • Axis of symmetry and vertex coordinates.

Example expressions

  • 1,-5,6
  • 1,2,1
  • 1,1,1
  • 2,0,-8
  • 3,7,-6
  • -1,4,5

Related formulas

  • Discriminant: D = b² − 4ac.
  • Roots: x = (−b ± √D)/(2a).
  • Axis of symmetry: x = −b/(2a).
  • Vertex: (−b/(2a), f(−b/(2a))).

How to use this calculator

  1. Read a, b and c from the equation in standard form.
  2. Enter all three coefficients, including zero when a term is missing.
  3. Calculate and inspect the discriminant, roots and graph details.

Common input mistakes

  • Coefficient a cannot be zero because that would make the equation linear.
  • Include the sign of each coefficient, for example b = -5 in x² - 5x + 6.
  • Enter c = 0 when the constant term is absent.

FAQ

What does the discriminant tell me?

A positive discriminant gives two real roots, zero gives one repeated root and a negative value gives two complex roots.

Does this calculator handle complex roots?

Yes. Negative discriminants are returned as a complex conjugate pair.

What if a is zero?

The equation is no longer quadratic, so this calculator asks for a non-zero value of a.

Does it graph the parabola?

It reports the vertex and axis of symmetry. An interactive graph is not included yet.

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