Completing the Square Calculator
How Completing the Square Works
Start with Standard Form
Begin with ax² + bx + c
Example: x² + 6x + 5
Find Half of b
Take b/2, then square it: (b/2)²
6/2 = 3, then 3² = 9
Add and Subtract
Add and subtract (b/2)² to complete the square
x² + 6x + 9 - 9 + 5
Factor Perfect Square
Group into (x ± h)² + k form
(x + 3)² - 4
Key Formula
Where:
h = -b/(2a)
k = c - b²/(4a)
Common Examples
Perfect Square Trinomials
Non-Perfect Squares
With Leading Coefficient
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
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Dr. Jane Doe
VerifiedExpert Reviewer & Mathematician
Last Updated: May 19, 2026