Math Calculator

Free Quadratic Equation Solver — Roots, Discriminant & Steps

Solve second-degree algebraic polynomials instantly with our free quadratic equation solver. Calculates real and complex roots with step-by-step math.

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Quadratic Equation Solver — Interactive Tool

Enter your values below for instant, high-precision results

Solve quadratic equation: ax² + bx + c = 0

What is the Quadratic Equation Solver?

The Quadratic Equation Solver from SmartCalc solves polynomial equations of the form ax² + bx + c = 0 using the quadratic formula. It calculates real and complex roots, displays the discriminant (Δ = b² − 4ac), identifies vertex coordinates, and reveals step-by-step algebraic steps.

How to Use the Quadratic Equation Solver — Step-by-Step

Follow these quick steps to calculate your results immediately:

  1. 1
    Identify the coefficients a, b, and c from your quadratic equation.
  2. 2
    Enter these values into the respective input fields.
  3. 3
    Click 'Solve Equation' to compute the roots.
  4. 4
    Review the calculated discriminant value and step-by-step solution pathway.

The Quadratic Formula

All calculations are computed based on mathematically rigorous standards and formulas:

Mathematical Formula x = (−b ± √(b² − 4ac)) ÷ (2a)  |  Discriminant Δ = b² − 4ac

Worked Example

Solving 2x² − 4x − 6 = 0 (a = 2, b = −4, c = −6):
1. Discriminant: (−4)² − 4(2)(−6) = 16 + 48 = 64
2. Square root: √64 = 8
3. Roots: x₁ = (4 + 8) / 4 = 3; x₂ = (4 − 8) / 4 = −1

Reference Data & Key Standards

Discriminant (Δ)Nature of RootsParabola Graph Behavior
Δ > 0Two distinct real rootsParabola intersects x-axis at two distinct points
Δ = 0One real repeated root (double root)Parabola vertex touches x-axis at exactly one point
Δ < 0Two complex conjugate roots (a ± bi)Parabola lies entirely above or below x-axis (no real intercepts)

Practical Tips & Best Practices

💡 Pro Tips:

  • Make sure your quadratic equation is arranged in standard form: ax² + bx + c = 0 before entering coefficients.
  • The coefficient 'a' cannot be zero; if a = 0, the equation is linear (bx + c = 0), not quadratic.
  • The vertex of the parabola always occurs at x = −b / (2a).

Frequently Asked Questions

Yes. If the discriminant (b² - 4ac) is negative, the quadratic solver automatically calculates and displays complex roots containing 'i'.

The discriminant determines the nature of the roots: positive means two real roots, zero means one repeated real root, and negative means two complex conjugate roots.

The equation must align with the standard quadratic formula form: ax² + bx + c = 0.

Key Takeaways
  • Solves any quadratic equation in real or complex number domains.
  • Calculates discriminant (Δ) and identifies root characteristics immediately.
  • Displays full step-by-step mathematical working for homework and algebra revision.
  • Zero registration or software installation required.