Quadratic Solver

Solve ax² + bx + c = 0 with discriminant and vertex.

1x² − 3x + 2 = 0

Discriminant

Nature

2 real roots

Vertex

(1.5, -0.25)

Roots

Two real roots:  x₁ = 2   x₂ = 1

Working

Δ = b² − 4ac = -3² − 4·1·2 = 1
x = (−b ± √Δ) / 2a
  = (3 ± 1) / 2

About the Quadratic Solver

Solve ax² + bx + c = 0 with full working shown. The tool computes the discriminant, classifies the roots as two real, one double or a complex conjugate pair, and reports the parabola vertex. Degenerate cases are handled too: when a is zero it solves the resulting linear equation instead.

How to use the Quadratic Solver

Enter coefficients a, b and c in the three fields.
Confirm the equation preview renders your values correctly.
Read the discriminant and nature tiles, then the Roots box.
Open the Working box to see the discriminant substitution and quadratic formula steps.

Quadratic Solver FAQ

What does the discriminant tell me?

Δ = b² − 4ac decides the root type: positive gives two distinct real roots, zero a single double root, and negative a complex pair written as x = re ± im·i.

What happens if a is zero?

The equation is no longer quadratic; the tool solves the linear remainder bx + c = 0, and warns outright if both a and b are zero.

Is the vertex computed as well?

Yes — the vertex (−b/2a, c − b²/4a) appears in its own tile, giving the parabola's turning point alongside the roots.