🧮 Completing the Square Calculator

📐 Completing the Square Calculator

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

ax² + bx + c = 0

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📝 Detailed Solution Steps

Enter a quadratic equation in the Calculator tab to see detailed steps here.

📊 Parabola Graph and Vertex

Enter a quadratic equation to see its graph and vertex information.

🎓 Completing the Square Theory

Completing the square is a method to convert a quadratic equation from standard form to vertex form, revealing the vertex of the parabola.

Standard Form: ax² + bx + c = 0
Vertex Form: a(x - h)² + k = 0
Where: (h, k) is the vertex of the parabola

General Method:

Step 1: Start with ax² + bx + c
Step 2: Factor out 'a' from x² and x terms: a(x² + (b/a)x) + c
Step 3: Complete the square inside parentheses: add and subtract (b/2a)²
Step 4: Rewrite as perfect square: a(x + b/2a)² + (c - b²/4a)
Step 5: Identify vertex: h = -b/2a, k = c - b²/4a

Example: x² + 6x + 5

Original: x² + 6x + 5
Step 1: Take half of coefficient of x: 6/2 = 3
Step 2: Square it: 3² = 9
Step 3: Add and subtract: x² + 6x + 9 - 9 + 5
Step 4: Group perfect square: (x + 3)² - 9 + 5
Result: (x + 3)² - 4
Vertex: (-3, -4)

Applications:

  • Finding Vertex: Identify the maximum or minimum point of a parabola
  • Solving Quadratics: Alternative to quadratic formula
  • Graphing: Easier to graph when in vertex form
  • Optimization: Find maximum/minimum values in real-world problems
  • Conic Sections: Standard form for parabolas in coordinate geometry

Key Formulas:

  • Vertex x-coordinate: h = -b/(2a)
  • Vertex y-coordinate: k = f(h) = ah² + bh + c
  • Discriminant: Δ = b² - 4ac
  • Axis of Symmetry: x = -b/(2a)