MultiConvers

Quadratic Formula Calculator

Enter coefficients a, b and c for the equation ax² + bx + c = 0. The tool reports the real roots, the discriminant and the vertex.

How it works

The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac decides how many real roots there are: two when Δ > 0, one repeated root when Δ = 0, and none when Δ < 0.

A negative discriminant is not printed as NaN — the page says there are no real roots and still shows Δ and the vertex (−b/2a, f(−b/2a)). a must not be 0; that would be a linear equation.

Formula

x = (−b ± √(b² − 4ac)) / (2a)
Δ = b² − 4ac
Vertex: (−b / 2a,  a(−b/2a)² + b(−b/2a) + c)

Worked examples

  • Two real roots

    a = 1, b = −5, c = 6 → x = 2 and x = 3

  • A repeated root

    a = 1, b = −2, c = 1 → x = 1, discriminant 0

Useful notes

  • Only real roots are listed. Complex conjugates are omitted on purpose.
  • The worked example x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0.

FAQ

What is the quadratic formula?

For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) / (2a), with a ≠ 0.

What does the discriminant tell me?

b² − 4ac. Positive: two distinct real roots. Zero: one real root. Negative: no real roots.

What if a is 0?

Then the equation is linear (bx + c = 0), not quadratic. This page rejects a = 0.

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