Completing the Square Calculator
How Completing the Square Works
1
Start with Standard Form
Begin with ax² + bx + c
Example: x² + 6x + 5
2
Find Half of b
Take b/2, then square it: (b/2)²
6/2 = 3, then 3² = 9
3
Add and Subtract
Add and subtract (b/2)² to complete the square
x² + 6x + 9 - 9 + 5
4
Factor Perfect Square
Group into (x ± h)² + k form
(x + 3)² - 4
Key Formula
ax² + bx + c
↓
a(x - h)² + k
Where:
h = -b/(2a)
k = c - b²/(4a)
Common Examples
Perfect Square Trinomials
x² + 6x + 9 → (x + 3)²
x² - 4x + 4 → (x - 2)²
x² + 10x + 25 → (x + 5)²
Non-Perfect Squares
x² + 6x + 5 → (x + 3)² - 4
x² - 4x + 3 → (x - 2)² - 1
x² + 8x + 12 → (x + 4)² - 4
With Leading Coefficient
2x² + 8x + 6 → 2(x + 2)² - 2
3x² - 12x + 9 → 3(x - 2)² - 3
-x² + 4x - 3 → -(x - 2)² + 1
a(x + h)² + k
Where h = b/(2a), k = c - b²/(4a)
Vertex Form
y = a(x-h)² + k
Parabola vertex
Quadratic Solving
ax² + bx + c = 0
Find roots
Optimization
Min/Max problems
Calculus applications
Graphing
Parabola analysis
Function behavior
What is Completing the Square?
📐
What
A method to rewrite quadratic expressions in the form a(x + h)² + k by creating a perfect square trinomial.
🎯
Why
Essential for solving quadratic equations, finding vertex of parabolas, and optimization problems in calculus.
⚡
Applications
Algebra (solving equations), geometry (parabola analysis), physics (projectile motion), and engineering optimization.
Step-by-Step Calculation Examples
| Original | Step 1: Identify a,b,c | Step 2: Calculate h,k | Final Form |
|---|---|---|---|
| x² + 6x + 5 | a=1, b=6, c=5 | h=-3, k=-4 | (x + 3)² - 4 |
| x² - 8x + 12 | a=1, b=-8, c=12 | h=4, k=-4 | (x - 4)² - 4 |
| 2x² + 12x + 10 | a=2, b=12, c=10 | h=-3, k=-8 | 2(x + 3)² - 8 |
| x² + 10x + 21 | a=1, b=10, c=21 | h=-5, k=-4 | (x + 5)² - 4 |
| -x² + 6x - 5 | a=-1, b=6, c=-5 | h=3, k=4 | -(x - 3)² + 4 |
Frequently Asked Questions
Reverse Conversion
unit to algebra-calculators →Popular Converters
Quick Reference
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