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Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0. Calculates real and complex roots, discriminant, vertex, and axis of symmetry.

1x²−5x +6= 0
Discriminant (Δ = b² − 4ac)1Two distinct real roots
Root x₁3
Root x₂2
Vertex (h, k)(2.5, -0.25)
Axis of Symmetryx = 2.5
Parabola OpensUpward ↑

Step-by-step Solution

  1. Identify: a = 1, b = -5, c = 6
  2. Compute discriminant: Δ = b² − 4ac = -5² − 4×1×6 = 1
  3. √Δ = √1 = 1
  4. x₁ = (−-5 + 1) / (2×1) = 3
  5. x₂ = (−-5 − 1) / (2×1) = 2
  6. Vertex: h = −b/(2a) = 2.5, k = -0.25

How to Use This Tool

  1. Enter the coefficient a (the number multiplying x²).
  2. Enter the coefficient b (the number multiplying x).
  3. Enter the constant c.
  4. Read the discriminant (Δ = b² − 4ac) to see whether the roots are real or complex.
  5. Review the calculated roots x₁ and x₂, along with the parabola's vertex and axis of symmetry.

Formula & How It Works

Quadratic Formula

x = (−b ± √(b² − 4ac)) / (2a)

Gives the two values of x that satisfy ax² + bx + c = 0.

Discriminant

Δ = b² − 4ac

If Δ > 0 there are two real roots, if Δ = 0 there is one repeated real root, and if Δ < 0 the roots are complex.

Vertex of the Parabola

h = −b / (2a), k = c − b² / (4a)

The point (h, k) is the highest or lowest point of the parabola, and x = h is its axis of symmetry.

Practical Examples & Common Use Cases

Example: x² − 5x + 6 = 0

a = 1, b = −5, c = 6. Δ = (−5)² − 4×1×6 = 25 − 24 = 1. x = (5 ± √1) / 2, giving x₁ = 3 and x₂ = 2.

Example: 2x² + 4x + 2 = 0 (repeated root)

a = 2, b = 4, c = 2. Δ = 4² − 4×2×2 = 16 − 16 = 0, so there is one repeated real root: x = −4 / (2×2) = −1.

Example: x² + x + 1 = 0 (complex roots)

a = 1, b = 1, c = 1. Δ = 1² − 4×1×1 = 1 − 4 = −3, which is negative, so the roots are complex: x = −0.5 ± 0.866i.

Frequently Asked Questions

x = (−b ± √(b² − 4ac)) / (2a). It gives the values of x where ax² + bx + c = 0.

The discriminant Δ = b² − 4ac tells us how many real solutions exist. Δ > 0: two real roots. Δ = 0: one repeated real root. Δ < 0: two complex (imaginary) roots.

The vertex is the highest or lowest point of the parabola. Its x-coordinate is h = −b/(2a) and y-coordinate is k = c − b²/(4a).

When a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula requires a ≠ 0.

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