Pure & Applied MathematicsEntry № 03.07Documented Formula

Quadratic Equation Solver with Step-by-Step Formula

Solves quadratic polynomials ax² + bx + c = 0, determining the discriminant, vertex coordinates, and real or complex roots with algebraic steps.

Popular Searches:How to solve quadratic equation formula: (-b ± √D)/2aParabola vertex coordinates (-b/2a, -D/4a)Quadratic equation with complex imaginary numbers

Quadratic Polynomial Algebraic Roots Solver

ax² + bx + c = 0

a (≠ 0)

1x² - 5x + 6 = 0

Discriminant: D = b² − 4ac = 1

Roots Solution

x₁ = 3.000, x₂ = 2.000

Nature: Two Distinct Real Roots

Parabola Vertex (h, k)(2.50, -0.25)

Parabola OpeningUpwards (Minimum Vertex)

Mathematical Proof & FormulaStandard Mathematical Notation

How This Calculation Formula is Formulated

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

Standard Algebraic Representation

x = [−b ± √(b² − 4ac)] / (2a) ; D = b² − 4ac

Documented algebraic formula with transparent derivation and reference notes.

Variable Definitions & Measurement Units

Quadratic Coefficient

Non-zero leading term (curvature).

Linear Coefficient

Slope at y-intercept.

Constant Term

Parabola y-intercept.

Discriminant

b² − 4ac determines root nature.

Step-by-Step Calculation Example

Solving x² − 5x + 6 = 0

Finding roots and vertex coordinates for a = 1, b = -5, c = 6.

Discriminant D

(-5)² − 4(1)(6) = 25 − 24 = 1

D > 0 indicates two distinct real roots

Root 1

[-(-5) + √1] / (2 × 1) = (5 + 1) / 2 = 3

First real root

Root 2

[-(-5) − √1] / (2 × 1) = (5 − 1) / 2 = 2

Second real root

Parabola Vertex

h = -(-5)/(2×1) = 2.5 ; k = -1/(4×1) = -0.25

Vertex at (2.5, -0.25)

Conclusion: The roots are x₁ = 3 and x₂ = 2 with parabola minimum vertex at (2.5, −0.25).

In-Depth Editorial Analysis

Algebraic Classification via the Discriminant D

When D > 0, the parabola intersects the x-axis at two distinct real coordinates. When D = 0, the vertex is tangent to the x-axis, creating a single repeated real root.

When D < 0, the parabola never crosses the x-axis in the Cartesian plane, producing two complex conjugate roots of the form u ± vi, crucial for electrical engineering and resonance damping equations.

Frequently Asked Questions

Frequently Asked Questions About Quadratic Equation Solver with Step-by-Step Formula

What happens if a = 0?

If a = 0, the equation reduces to a linear equation bx + c = 0 with a single root x = -c/b.

What is the axis of symmetry of a parabola?

The vertical line passing through the vertex, given by x = -b / (2a).

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Deterministic Precision & 100% Client-Side Privacy

All calculations execute in your local browser using IEEE 754 double-precision floating-point arithmetic. Your figures and financial metrics remain private and are never uploaded or saved to any cloud servers.