Completing the Square Calculator

Solve quadratic equations by completing the square method. This calculator provides step-by-step solutions for equations in the form ax² + bx + c = 0.

Equation:

1x² + 6x + 5 = 0

How Completing the Square Works

Completing the square is a method for solving quadratic equations by creating a perfect square trinomial. The process involves these main steps:

1. If a ≠ 1, divide equation by a

2. Move constant term to right side

3. Add (b/2a)² to both sides

4. Factor left side as perfect square

5. Solve for x using square root

Example Solutions

Simple Example

x² + 6x + 5 = 0

  1. Move constant: x² + 6x = -5
  2. Add (b/2)² = 9: x² + 6x + 9 = -5 + 9
  3. Factor: (x + 3)² = 4
  4. Solve: x + 3 = ±2
  5. Therefore: x = -1 or x = -5

Complex Example

2x² - 12x + 10 = 0

  1. Divide by 2: x² - 6x + 5 = 0
  2. Move constant: x² - 6x = -5
  3. Add (b/2)² = 9: x² - 6x + 9 = -5 + 9
  4. Factor: (x - 3)² = 4
  5. Solve: x - 3 = ±2
  6. Therefore: x = 5 or x = 1

Real-World Applications

Physics: Projectile Motion

A ball is thrown upward with initial velocity of 20 m/s from a height of 1.5m.

h = -4.9t² + 20t + 1.5

Find when ball returns to ground:

-4.9t² + 20t + 1.5 = 0

Solution: t ≈ 4.2 seconds

Engineering: Bridge Design

The shape of a suspension cable follows a quadratic equation for a 100m wide bridge:

y = 0.004x² - 0.4x + 10

Complete the square to find lowest point:

y = 0.004(x² - 100x) + 10

y = 0.004(x - 50)² + 0

Lowest point at x = 50m

Economics: Profit Optimization

Find the price that maximizes profit using the profit function:

P = -2x² + 120x - 1000

Complete square to find maximum profit:

P = -2(x² - 60x) - 1000

P = -2(x - 30)² + 800

Maximum profit at x = $30

Architecture: Arch Design

Calculate the maximum height of a Roman arch using its profile equation:

h = -0.1x² + 1.5x + 6

Find maximum height:

h = -0.1(x² - 15x) + 6

h = -0.1(x - 7.5)² + 11.625

Maximum height at x = 7.5m

Industry Applications

Manufacturing

  • Optimizing container volumes
  • Minimizing material waste
  • Production rate optimization
  • Quality control tolerances

Finance

  • Break-even analysis
  • Investment return modeling
  • Risk assessment curves
  • Portfolio optimization

When to Use This Method

  • When you need to see the steps of the solution
  • When the quadratic formula is too complex
  • When you need to find the vertex of a parabola
  • When working with perfect square trinomials
  • When solving equations in standard form (ax² + bx + c = 0)

Solving: 2x² + 4x + 5 = 0

2x² + 4x + 5 = 0

1. First divide everything by 2:

x² + 2x + 2.5 = 0

2. Add (b/2)² = 1 to both sides:

x² + 2x + 1 = -2.5 + 1

3. Factor the perfect square:

(x + 1)² = -1.5

Since the square of a real number cannot be negative, this equation has no real solutions.

Tips for Success

  • Always check if a = 1; if not, divide the equation by a first
  • The term (b/2a)² is key to completing the square
  • Remember to perform the same operations on both sides
  • Check your answer by substituting back into the original equation
  • Watch for negative numbers when taking square roots