Quadratic Formula Calculator
Solve quadratic equations ax² + bx + c = 0, find real and complex roots, discriminant, and parabola vertex.
Input Parameters
Live Result
Details
Steps
Calculate Discriminant: Δ = b² - 4ac
Apply Quadratic Formula: x = (-b ± √Δ) ÷ (2a)
Compute Parabola Vertex: x_v = -b ÷ 2a
Quick Answer & Understanding Your Result
A Quadratic Formula Calculator solves second-degree polynomial equations (ax^2 + bx + c = 0), determining real or complex roots, discriminant (Delta), and parabola vertex coordinates.
Discriminant Analysis Criteria (Delta = b^2 - 4ac)
- Delta > 0: Two distinct real roots (x_{1,2} = (-b pm sqrt{Delta}) / 2a).
- Delta = 0: One real double root / vertex tangency (x = -b / 2a).
- Delta < 0: Two complex conjugate roots (x = (-b pm isqrt{|Delta|}) / 2a).
- Parabola Vertex: The extremum coordinate at (h, k) = (-b / 2a, c - b^2 / 4a).
Physical & Engineering Uses
- Projectile Trajectory & Ballistics: Computing maximum flight apex height and impact landing times.
- Profit & Cost Optimization: Determining maximum profit vertices for parabolic economic models.
Enter coefficients a, b, and c above to solve roots and view the step-by-step quadratic breakdown instantly.
Formula & Calculation Logic
Mathematical FormulaQuadratic derivation:
- Solves second-order algebraic equation through completing the square.
- a
eq 0 ensures a true quadratic parabola.
Step-by-Step Guide
Enter Coefficient a
Input quadratic coefficient $a$ ($a eq 0$).
Enter Coefficients b and c
Input linear coefficient $b$ and constant term $c$.
View Roots & Vertex
Receive real/complex solutions, discriminant value, and step-by-step derivation.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Standard Quadratic: x² - 5x + 6 = 0 | Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. x = (5 ± 1) / 2 → x₁ = 3, x₂ = 2. | ||
| Double Root: x² - 4x + 4 = 0 (Root: 2) | Provides instant quantitative evaluation for comparative practical analysis. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
NIST Digital Library of Mathematical Functions - Algebraic Equations
Khan Academy - Quadratic Equations and Functions
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: x = (-b ± √(b² - 4ac)) ÷ (2a)
Last Reviewed: August 21, 2026
Primary Source: NIST Digital Library of Mathematical Functions - Algebraic Equations
Reference Document: Khan Academy - Quadratic Equations and Functions
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.