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Completing the Square Calculator

Solve quadratic equations using the completing the square method. Get step-by-step solutions, vertex coordinates, and discriminant analysis for any quadratic equation in standard form.

Complete the Square Calculator

Enter coefficients for ax² + bx + c = 0

Coefficient a (x² term)

Must not be zero

Coefficient b (x term)
Constant c

Completing the Square Results

Original Equation

x² + 6x + 5 = 0

Completed Square Form

(x - 3)² - 4

Vertex Information

Vertex: (-3.000, -4.000)
Axis of Symmetry: x = -3.000
Opens: Upward
Y-intercept: (0, 5)

Solutions

Discriminant: 16.000 (Two real solutions)
x = -5.000
x = -1.000

Step-by-Step Process

1Original equation: x² + 6x + 5 = 0
2Standard form: 1x² + 6x + 5 = 0
3Step 1: Factor out coefficient of x²: 1(x² + 6x) + 5
4Step 2: Complete the square: 1(x² + 6x + 9) + 5 - 1 × 9
5Step 3: Simplify: 1(x + 3)² + -4

Types of Quadratic Equations

Standard Forms

Standard Formax² + bx + c = 0

Most common form for solving

  • • Easy to identify coefficients
  • • Direct application of formulas
Vertex Forma(x - h)² + k = 0

Result of completing the square

  • • Vertex at (h, k)
  • • Easy to graph
Factored Forma(x - r)(x - s) = 0

When equation can be factored

  • • Solutions are r and s
  • • X-intercepts visible

Solution Types

Two Real SolutionsΔ > 0

Discriminant positive

  • • Parabola crosses x-axis twice
  • • Two distinct real roots
One Real SolutionΔ = 0

Discriminant zero

  • • Parabola touches x-axis once
  • • Perfect square trinomial
No Real SolutionsΔ < 0

Discriminant negative

  • • Parabola doesn't cross x-axis
  • • Complex solutions exist

How to Complete the Square

Completing the Square Formula

Step 1: Start with ax² + bx + c = 0
Step 2: Factor out a: a(x² + (b/a)x) + c = 0
Step 3: Add (b/2a)²: a(x² + (b/a)x + (b/2a)²) + c - a(b/2a)² = 0
Step 4: Factor: a(x + b/2a)² + c - b²/4a = 0

Source: Standard algebraic technique taught in high school mathematics curriculum

Step-by-Step Process:

  1. 1
    Ensure coefficient of x² is 1
    If a ≠ 1, factor it out from the x² and x terms
  2. 2
    Move constant to the other side
    Isolate the x² and x terms on one side
  3. 3
    Add (b/2)² to both sides
    This creates a perfect square trinomial
  4. 4
    Factor the perfect square
    Write as (x + b/2)²
  5. 5
    Solve for x
    Take square root and isolate x

Important Considerations

⚠️ Mathematical Accuracy

This calculator provides precise algebraic solutions. Always verify results for critical applications.

🔢 Coefficient Requirements

The coefficient 'a' must not be zero

  • • If a = 0, equation becomes linear
  • • Use different methods for linear equations
  • • Quadratic formula requires a ≠ 0
📐 Discriminant Analysis

The discriminant determines solution types

  • • Δ = b² - 4ac
  • • Positive: two real solutions
  • • Zero: one real solution
  • • Negative: complex solutions
📊 Graphing Applications

Vertex form makes graphing easier

  • • Vertex at (h, k)
  • • Axis of symmetry: x = h
  • • Direction: up if a > 0, down if a < 0
⚠️ Common Mistakes

Avoid these common algebraic errors

  • • Forgetting to add (b/2a)² to both sides
  • • Incorrect sign when taking square root
  • • Not factoring out 'a' when a ≠ 1

Example Cases

Case 1: Simple Quadratic (x² + 6x + 5 = 0)

Input: a = 1, b = 6, c = 5
Process: Complete the square
Result: (x + 3)² - 4 = 0
Vertex: (-3, -4)
Solutions: x = -1, x = -5
Discriminant: 16 (two real solutions)

Use Case: Perfect for learning the basic technique with integer coefficients and clear solutions.

Case 2: Complex Quadratic (2x² - 8x + 3 = 0)

Input: a = 2, b = -8, c = 3
Process: Factor out 2, complete square
Result: 2(x - 2)² - 5 = 0
Vertex: (2, -5)
Solutions: x ≈ 3.58, x ≈ 0.42
Discriminant: 40 (two real solutions)

Use Case: Demonstrates completing the square with a ≠ 1 and irrational solutions.

Case 3: Perfect Square (x² - 4x + 4 = 0)

Input: a = 1, b = -4, c = 4
Process: Already a perfect square
Result: (x - 2)² = 0
Vertex: (2, 0)
Solutions: x = 2 (double root)
Discriminant: 0 (one real solution)

Use Case: Shows the special case where the discriminant is zero and there's exactly one solution.

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Frequently Asked Questions

What is completing the square method?
Completing the square is an algebraic technique used to solve quadratic equations and convert them into vertex form. It involves adding and subtracting a constant term to create a perfect square trinomial, making it easier to solve or graph the equation.
When should I use completing the square?
Use completing the square when you need to find the vertex of a parabola, solve quadratic equations that don't factor easily, or convert quadratic equations to vertex form. It's particularly useful for graphing and understanding the properties of quadratic functions.
What are the steps in completing the square?
The basic steps are: 1) Ensure the coefficient of x² is 1, 2) Move the constant term to the other side, 3) Add (b/2)² to both sides, 4) Factor the perfect square trinomial, 5) Solve for x. Our calculator shows each step in detail.
What is vertex form and why is it useful?
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex, axis of symmetry, and direction of opening, which are crucial for graphing and understanding quadratic functions.
What does the discriminant tell us?
The discriminant (b² - 4ac) determines the nature of solutions: positive means two real solutions, zero means one real solution, and negative means no real solutions (complex solutions). It also indicates whether the parabola intersects the x-axis.
Completing the Square Calculator