Completing the Square Calculator
Result:
Master the Completing the Square technique with our comprehensive calculator. Transform any quadratic expression from standard form to vertex form, find vertices of parabolas, and solve quadratic equations efficiently.
What is Completing the Square?
Completing the square is an algebraic technique used to convert a quadratic expression from standard form (ax² + bx + c) to vertex form a(x - h)² + k, revealing the vertex (h, k) of the parabola.
The Completing the Square Process
Step-by-Step Method
Starting with: ax² + bx + c
- Factor out 'a': a(x² + (b/a)x) + c
- Complete the square inside parentheses: Take half of (b/a), then square it
- Add and subtract: a(x² + (b/a)x + (b/2a)² - (b/2a)²) + c
- Factor the perfect square: a((x + b/2a)² - (b/2a)²) + c
- Distribute and simplify: a(x + b/2a)² - b²/4a + c
- Final form: a(x - h)² + k where h = -b/2a
Detailed Example: Complete x² + 6x + 5
Step-by-Step Solution
Given: x² + 6x + 5
Step 1: Identify coefficients
a = 1, b = 6, c = 5
Step 2: Take half of b coefficient and square it
(6/2)² = 3² = 9
Step 3: Add and subtract this value
x² + 6x + 9 - 9 + 5
Step 4: Factor the perfect square trinomial
(x + 3)² - 9 + 5
Step 5: Simplify
(x + 3)² - 4
Result: Vertex form is (x + 3)² - 4 with vertex (-3, -4)
Forms of Quadratic Expressions
Standard Form
ax² + bx + c
Example: 2x² + 8x + 6
Shows: y-intercept (c)
Use: Finding intercepts
Vertex Form
a(x - h)² + k
Example: 2(x - 2)² - 2
Shows: Vertex (h, k)
Use: Graphing parabolas
Factored Form
a(x - r₁)(x - r₂)
Example: 2(x - 1)(x - 3)
Shows: x-intercepts (r₁, r₂)
Use: Finding roots
Applications of Completing the Square
- Solving quadratic equations: Alternative to quadratic formula
- Graphing parabolas: Easily identify vertex and axis of symmetry
- Optimization problems: Find maximum or minimum values
- Conic sections: Converting equations to standard form
- Calculus integration: Simplifying integrals involving quadratics
- Physics applications: Projectile motion and optimization
Special Cases and Tips
When a ≠ 1
Example: 2x² + 8x + 6
Step 1: Factor out 2: 2(x² + 4x) + 6
Step 2: Complete inside: 2(x² + 4x + 4 - 4) + 6
Step 3: Factor: 2((x + 2)² - 4) + 6
Step 4: Distribute: 2(x + 2)² - 8 + 6
Result: 2(x + 2)² - 2
Perfect Square Recognition
Pattern: x² ± 2bx + b²
Factors to: (x ± b)²
Examples:
- x² + 6x + 9 = (x + 3)²
- x² - 10x + 25 = (x - 5)²
- x² + 14x + 49 = (x + 7)²
Connection to the Quadratic Formula
Completing the square is actually how the quadratic formula is derived! When you complete the square for the general equation ax² + bx + c = 0, you get:
x = (-b ± √(b² - 4ac)) / 2a
Practice Problems
Basic Example
Complete: x² + 4x + 1
Answer: (x + 2)² - 3
Vertex: (-2, -3)Leading Coefficient ≠ 1
Complete: 3x² + 12x + 7
Answer: 3(x + 2)² - 5
Vertex: (-2, -5)Negative Coefficient
Complete: -x² + 6x - 5
Answer: -(x - 3)² + 4
Vertex: (3, 4)Vertex Information
Once in vertex form a(x - h)² + k, you can immediately identify:
- Vertex coordinates: (h, k)
- Axis of symmetry: x = h
- Direction of opening: Up if a > 0, Down if a < 0
- Maximum/minimum value: k (minimum if a > 0, maximum if a < 0)
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