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Quadratic Equation Solver

Solve quadratic equations ax² + bx + c = 0 with step-by-step discriminant derivations and vertex formulas.

ax² + bx + c = 0
Roots / Solutions2 Real Roots
Root x₁

3

Root x₂

2

Discriminant (Δ = b² - 4ac)1
Vertex Coordinates (h, k)(2.5, -0.25)
Step-by-Step Derivation Breakdown
1. Standard Form: (1)x² + (-5)x + (6) = 0
2. Discriminant: Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
3. Since Δ > 0, there are two distinct real roots.
4. Quadratic Formula: x = (-b ± √Δ) / 2a = (-(-5) ± √1) / 2(1)
5. Vertex: (h, k) = (-b/2a, c - b²/4a) = (2.50, -0.25)
6. Parabola opens: Upward (Minimum at vertex)

How to Use the Quadratic Equation Solver

1

Enter the coefficients a, b, and constant c in standard quadratic form.

2

View the computed discriminant (Δ), real or complex roots (x₁, x₂), and vertex coordinates.

3

Read through the full step-by-step algebraic derivation breakdown.

Key Capabilities & Features

  • •Solves 2 real distinct roots, repeated roots, and complex conjugate numbers
  • •Calculates discriminant value (Δ = b² - 4ac) and parabola vertex (h, k)
  • •Complete step-by-step algebraic derivation explanations
Input FormatsCoefficients a, b, c
Output FormatsRoots x1, x2, Discriminant, Vertex Coordinates, Step-by-step Derivation

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