MultiConvers

Quadratic Formula Calculator

Solve ax² + bx + c = 0 on the real line and see why: discriminant, roots and the parabola’s vertex — not complex conjugates and not a linear bx + c = 0.

How to use

  1. Enter a, b and c from the standard form. a must not be 0.
  2. Read the discriminant first: positive two roots, zero one root, negative none real.
  3. Read the real roots and the vertex (−b/2a, f of that x).
  4. If a is 0, solve bx + c = 0 by hand; this page will refuse the input.

How it works

The quadratic formula is x = (−b ± √Δ) / (2a) with Δ = b² − 4ac. x² − 5x + 6 = 0 has Δ = 25 − 24 = 1, roots 2 and 3, which match the factorisation (x − 2)(x − 3). x² − 2x + 1 = 0 has Δ = 0 and a repeated root 1. x² + 1 = 0 has Δ = −4 and no real roots; the page says so instead of printing NaN.

The vertex is the turning point of y = ax² + bx + c. Its x-coordinate is −b/(2a). For y = x² − 5x + 6 that is 2.5, and y(2.5) = −0.25. A positive a opens up; a negative a opens down.

Complex roots come in conjugates when Δ < 0. This tool omits them on purpose. Completing the square is the same algebra in disguise; you do not need it to read the result.

Units are numbers, not metres. A projectile “height” model is only as good as the coefficients you typed.

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

  • Real roots only. Negative Δ is “no real roots,” not NaN.
  • a = 0 is linear, not quadratic.
  • x² − 5x + 6 = 0 → x = 2 and x = 3.

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.

What are the roots of x² − 5x + 6 = 0?

2 and 3. Discriminant 1, vertex at x = 2.5.

Where is the vertex of y = x² − 4x + 3?

x = 2, y = −1. Roots 1 and 3 sit symmetrically around that line.

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