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Math Solvers/Algebra & Calculus/quadratic-polynomial

Quadratic Equation & Parabola Vertex Solver

Solve any quadratic polynomial ax² + bx + c = 0 with comprehensive KaTeX steps, discriminant analysis, vertex coordinate determination, and parabolic properties.

Polynomial Coefficients

Current Equation
First Root (x₁)
3.0000
Real intersection root
Second Root (x₂)
2.0000
Secondary root
Discriminant (Δ)
1.0000
2 Real Roots (Δ > 0)
Parabola Vertex (h, k)
(2.500, -0.250)
Axis of symmetry: x = 2.500

KaTeX Step-by-Step Derivation

1. Quadratic Equation Form & Coefficients

Identified coefficients: a = 1, b = -5, c = 6.

2. Discriminant Evaluation (Δ)

Δ > 0: The equation has two distinct real roots.

3. Quadratic Formula Solution

Calculated roots: x_1 = 3.0000, x_2 = 2.0000.

4. Parabola Vertex & Geometry

The axis of symmetry is x = 2.5000. Parabola opens upwards (minimum point).

Recommended Handbooks & Professional Tools

Essential high-yield reference standards and hardware for Quadratic Equation & Parabola Vertex Solver

Mechanical & ManufacturingIndustry Bible ($120+)

Machinery's Handbook (31st Edition - Toolbox Edition)

by Erik Oberg, Franklin D. Jones

The 3,000-page ultimate manufacturing & mechanical engineering reference.

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Civil & StructuralEssential Code ($300+)

AISC Steel Construction Manual (16th Edition)

by American Institute of Steel Construction

The official code manual for structural steel design, member tables & specifications.

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Mathematical Formulation

Standard analytical and numerical formulations derived from classical algebra, linear algebra, and calculus principles.

Engineering Reference & Technical Guide

Quadratic Equation & Parabola Analysis Reference Guide

Peer-reviewed mathematical formulation, verified step-by-step worked engineering calculations, design codes, and practical guidelines.

Governing Theory & Derivation

Analytical physics and classical mechanics foundations

A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. It models trajectories in projectile motion, structural arch profiles, profit optimization, and kinetic energy relationships.

1. Completing the Square Derivation

Dividing through by 'a' and completing the square produces the universal quadratic formula:

2. Discriminant Analysis (Δ)

The discriminant Δ = b² - 4ac dictates root characteristics: If Δ > 0, there are two distinct real roots; if Δ = 0, there is a single repeated real root; if Δ < 0, there are two complex conjugate roots:

3. Parabola Vertex and Extremum

The vertex represents the global minimum (if a > 0) or maximum (if a < 0) along the axis of symmetry:

Engineering Interpretation:In structural mechanics and kinematics, the vertex of a parabolic bending moment diagram represents the location of maximum bending moment where shear force vanishes.

Step-by-Step Worked Example

Civil Engineering Parabolic Suspension Bridge Cable Sag

Problem Statement

A suspension bridge cable follows the parabolic profile y = 0.002x² - 0.24x + 10 over a 120-meter span (from x = 0 to x = 120). Determine the sag at the lowest point (vertex) and the span coordinates where the cable reaches an elevation of y = 6.0 meters.

Given Parameters

Quadratic Coefficient (a)0.002 m^-1
Linear Coefficient (b)-0.24 dimensionless
Constant Coefficient (c)10.0 m

Detailed Calculation Steps

1Find Parabola Vertex (Lowest Point of Cable)

The lowest cable point is located at mid-span (x = 60 m) with an elevation of 2.80 meters.

2Solve for Coordinates where y = 6.0 m

Setting equation equal to target elevation y = 6.0 m.

3Apply Quadratic Formula

The cable reaches 6.0 m elevation symmetrically at x = 20 m and x = 100 m.

Conclusion & Design Verification:The cable profile sags to a minimum height of 2.80 meters at mid-span, clearing the roadway structure with symmetric hanger anchor points at x = 20 m and 100 m.

Applicable Standards & Design Codes

Recognized engineering guidelines and criteria

AASHTO LRFDAASHTO

Bridge Design Specifications

Guidelines for parabolic cable geometries and roadway vertical curves.

Assumptions & Practical Limitations

Critical constraints for professional applications

  • Suspension cables under self-weight follow a catenary curve (cosh). A parabolic approximation is valid only when sag-to-span ratio is less than 1:10.

Frequently Asked Questions

Common technical engineering inquiries

Q:When are quadratic roots complex?

Roots are complex numbers when the discriminant Δ = b² - 4ac is negative. Geometrically, this means the parabola never crosses the horizontal x-axis.