Math Toolquadratic equation calculatorUpdated 2026

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

Find real and complex roots of quadratic equations with step-by-step discriminant evaluation and parabola vertex coordinates.

Coefficients (ax² + bx + c = 0)

Roots of the Equation
x₁ = 3
x₂ = 2
Two Distinct Real Roots
Discriminant (b² - 4ac)
1
Parabola Vertex (h, k)
(2.50, -0.25)
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How Quadratic Equation Works: Mathematical Formula & Steps

Solves second-degree polynomial equations of the form ax² + bx + c = 0, determining real, repeated, or complex conjugate roots and parabola vertex coordinates.

Quadratic Formula & Discriminant Analysis
x = [ −b ± √(b² − 4ac) ] / (2a) | Discriminant (Δ) = b² − 4ac
Where:
  • a (Quadratic Coefficient): Coefficient of x² (cannot equal 0).
  • b (Linear Coefficient): Coefficient of x.
  • c (Constant Term): Y-intercept of the parabola.
  • Δ (Discriminant): If Δ > 0, 2 real roots; if Δ = 0, 1 real root; if Δ < 0, 2 complex roots.

Worked Real-World Example: Step-by-Step Calculation

To illustrate how the quadratic equation calculator works in practice, consider the following real scenario:

Solving x² − 5x + 6 = 0

Scenario: a = 1, b = −5, c = 6.

Step 1: Evaluate discriminant Δ = b² − 4ac
(−5)² − 4(1)(6) = 25 − 24 = 1
Step 2: Apply quadratic formula
x = [ −(−5) ± √1 ] / (2 × 1) = [ 5 ± 1 ] / 2
Step 3: Solve for x₁ and x₂
x₁ = (5+1)/2 = 3, x₂ = (5−1)/2 = 2
Final Result: The roots are real integers: x = 3 and x = 2. Vertex is at (2.5, −0.25).

Key Takeaways & Best Practices

  • Vertex coordinates (h, k) are located at h = −b / (2a) and k = c − (b² / 4a).

International & Multi-Currency Standards

Whether you are evaluating calculations in the United States (USD), United Kingdom (GBP), Eurozone (EUR), South Africa (ZAR), Australia (AUD), Canada (CAD), or India (INR), everycalculator dynamically formats outputs according to regional accounting notations and dual metric/imperial conversions.

Related Search Terms & Common Queries

This calculator is designed to provide immediate answers for popular search queries including:

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Frequently Asked Questions

What does a negative discriminant mean?

When the discriminant (b² − 4ac) is less than zero, the quadratic equation has no real solutions (the parabola does not cross the x-axis). Instead, it has two complex/imaginary roots containing the imaginary unit i (where i = √−1).

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