Completing the Square Calculator
Solve quadratic equations and convert to vertex form using the completing the square method
Complete the Square
Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0:
Cannot be zero
Can be any real number
Can be any real number
Your quadratic equation:
Results
Vertex Form
Vertex: (0.000, 0.000)
Discriminant
b² - 4ac = 0² - 4(1)(0)
Roots/Solutions
Step-by-Step Solution
Step 1: Original Equation
Start with the quadratic equation:
Step 2: Move Constant
Move the constant term to the right side:
Step 3: Find Completing Value
Take half of the coefficient of x and square it: (0/2)² = 0
Step 4: Add to Both Sides
Add 0 to both sides:
Step 5: Factor Perfect Square
Factor the left side as a perfect square:
Step 6: Take Square Root
Take the square root of both sides:
Step 7: Solve for x
Solve for x:
Example Problems
Example 1
Example 2
Example 3
Example 4
Completing the Square Steps
Key Formulas
Tips & Notes
If a ≠ 1, divide entire equation by a first
Vertex form reveals the parabola's vertex directly
If b = 0, no need to complete the square
Always verify by expanding back to standard form
Applications
Understanding Completing the Square
What is Completing the Square?
Completing the square is a method used to solve quadratic equations and convert them from standard form (ax² + bx + c = 0) to vertex form (a(x - h)² + k = 0). This technique "completes" a partial perfect square trinomial by adding the appropriate constant.
Why Use This Method?
- •Reveals the vertex of the parabola directly
- •Makes graphing parabolas easier
- •Useful for optimization problems
- •Foundation for deriving the quadratic formula
The Perfect Square Pattern
The key insight is recognizing the pattern of perfect square trinomials:
When we have x² + bx, we need to add (b/2)² to complete the square pattern.
Geometric Interpretation
Geometrically, completing the square can be visualized as literally completing a square shape by adding the missing corner piece. The area x² + bx can be arranged as a rectangle, and we add (b/2)² to make it a perfect square.
Connection to the Quadratic Formula
Completing the square for the general equation ax² + bx + c = 0 actually gives us the quadratic formula:
This formula is derived by completing the square on the general quadratic equation.