Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0. Calculate roots (real & complex), discriminant, vertex coordinates, axis of symmetry, and step-by-step factorization.

Equation Coefficients

x² -5x + 6 = 0
Quick Presets
Quadratic Roots (x-intercepts)
x₁ =3
x₂ =2

Factored: (x - 3)(x - 2) = 0

2 Real Roots (Δ > 0)

Discriminant (Δ)

1

b² − 4ac

Vertex (h, k)

(2.5, -0.25)

extremum point

Axis of Symmetry

x = 2.5

vertical axis

Y-Intercept

(0, 6)

at x = 0
Parabola Geometric Properties
Opening Direction:Opens Upward (∪ Min)
a > 0
Root Nature:2 Real Distinct Roots
Δ = 1
Exact Step-by-Step Solution
Step 1: Calculate Discriminant ΔΔ = (-5)² − 4(1)(6) = 25 − 24 = 1
Step 2: Substitute into Quadratic Formulax = [−(-5) ± √(1)] / [2(1)]
Step 3: Solve for xx₁ = 3, x₂ = 2
Standard Quadratic Equations & Properties
Quadratic Formula
x =
−b ± √(b² − 4ac)2a
Discriminant (Δ)Δ = b² − 4ac
Parabola Vertex (h, k)h = −b / (2a)
Step-by-Step Calculation Breakdown
Example: Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6)
1. Discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
2. Numerator: −(−5) ± √1 = 5 ± 1
3. Denominator: 2(1) = 2
4. Root 1: x₁ = (5 + 1) / 2 = 3
5. Root 2: x₂ = (5 − 1) / 2 = 2
Factored Form:(x − 3)(x − 2) = 0

Key Parabola & Algebra Concepts

  • Two Real Roots (Δ > 0): The parabola intersects the x-axis at two unique coordinates.
  • Single Double Root (Δ = 0): The parabola is tangent to the x-axis at its vertex point.
  • Complex Roots (Δ < 0): The parabola is suspended entirely above or below the x-axis, yielding imaginary roots ± i.

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It provides the exact analytical solutions (roots / x-intercepts) for any polynomial equation of degree 2 in standard form ax² + bx + c = 0.
What does the discriminant (Δ = b² − 4ac) indicate?
If Δ > 0, the equation has two distinct real roots (crosses the x-axis twice). If Δ = 0, it has one real double root (parabola touches the x-axis at its vertex). If Δ < 0, it produces two complex conjugate solutions (the parabola never intersects the x-axis).
What is the vertex and axis of symmetry of a parabola?
The axis of symmetry is the vertical line x = −b / (2a). The vertex is the parabola's extremum point located at (x_v, y_v) = (−b/(2a), f(−b/(2a))). If a > 0, the vertex is a global minimum; if a < 0, it is a global maximum.
When can a quadratic equation be factored easily?
A quadratic equation can be factored cleanly over integers whenever the discriminant Δ is a perfect square (e.g. 1, 4, 9, 16, 25, 36, ...), giving rational integer roots.

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