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100% Client-Side Quadratic calculations run in local browser RAM.

Quadratic Equation Solver (ax² + bx + c = 0)

Calculate real and complex roots, discriminant, parabola vertex, and step-by-step quadratic formula

Algebra & Matrix Engine

Algebra & Matrix Studio

Solve quadratic equations with complex roots, 2x2 systems via Cramer's rule, binomial FOIL expansions, and 2x2/3x3 matrix linear algebra with instant proofs.

Quadratic Coefficients

Standard form: ax² + bx + c = 0 (where a ≠ 0).
Sample Presets:

Parabola Geometry

Vertex (h, k)(2.5, -0.25)
Axis of Symmetryx = 2.5
OrientationOpens Upward (Minimum)
Discriminant (Δ)1
Two Distinct Real Roots

1x² - 5x + 6 = 0

Root x1
3
Root x2
2
Factored Form: (x - 3)(x - 2) = 0

Step-by-Step Derivation (Quadratic Formula)

1Equation: 1x² - 5x + 6 = 0
2Discriminant Δ = b² - 4ac = (-5)² - 4(1)(6) = 1
3Δ > 0: Two distinct real roots.
4x₁ = (-(-5) + √1) / (2 × 1) = 3
5x₂ = (-(-5) - √1) / (2 × 1) = 2
6Parabola Vertex (h, k) = (-b/2a, c - b²/4a) = (2.5, -0.25)

About Quadratic Equation Solver (ax² + bx + c = 0)

Solve quadratic equations ax² + bx + c = 0 with precision. Computes discriminant Δ, real and complex roots x₁ and x₂, parabola vertex (h, k), axis of symmetry, and full quadratic formula derivations.

Key Capabilities & Features

  • Discriminant Δ = b² - 4ac computation
  • Real and complex conjugate roots (u ± vi)
  • Parabola vertex (h, k) and concavity direction
  • Complete step-by-step formula derivation

How to Use Quadratic Equation Solver (ax² + bx + c = 0)

1

Enter Coefficients

Input numerical values for a, b, and c.

2

Check Discriminant

Determine if roots are real or complex.

3

Inspect Roots

Review exact solutions and parabola vertex coordinates.

Privacy & In-Browser Execution Guarantee

Quadratic calculations run in local browser RAM.

Frequently Asked Questions

What does a negative discriminant mean?

When Δ < 0, the quadratic equation has no real roots and produces two complex conjugate solutions involving imaginary unit i.

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