Advanced Completing the Square Calculator

Advanced Completing the Square Calculator

Transform quadratic equations to vertex form with step-by-step solutions

📐 Quadratic Equation

Enter coefficients for ax² + bx + c = 0

x² + 6x + 8 = 0

📊 Solution Results

Completed square form and analysis

Original: x² + 6x + 8 = 0
Completed Square: (x + 3)² - 1 = 0
Vertex: (-3, -1)
Solutions: x = -2, -4
Discriminant: 4

📈 Parabola Graph

Visual representation of the quadratic function

🎯 3D Parabola

Interactive 3D visualization

Mouse: Rotate view
Wheel: Zoom in/out

Completing the Square Guide

Completing the square is a method for solving quadratic equations and converting them to vertex form. This technique reveals the vertex of the parabola and makes solving easier.

Completing the Square Process

1. Start with ax² + bx + c = 0
2. Factor out 'a': a(x² + (b/a)x) + c = 0
3. Complete: a(x² + (b/a)x + (b/2a)²) - a(b/2a)² + c = 0
4. Simplify: a(x + b/2a)² + (c - b²/4a) = 0

Vertex Form: The completed square form a(x - h)² + k = 0 immediately reveals the vertex (h, k) of the parabola, making graphing and analysis straightforward.

Applications: This method is essential in calculus for optimization problems, physics for projectile motion, and engineering for modeling parabolic relationships.

Discriminant: The value b² - 4ac determines the nature of solutions: positive (two real roots), zero (one repeated root), or negative (complex roots).

Our Website