1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by converting them into perfect square trinomials. This technique is fundamental in algebra and provides insight into the nature of quadratic solutions.

2. How the Calculator Works

The calculator solves quadratic equations of the form:

It follows these steps:

1. Divides all terms by a (if a ≠ 1)
2. Moves the constant term to the right side
3. Adds the square of half the coefficient of x to both sides
4. Writes the left side as a perfect square
5. Takes the square root of both sides
6. Solves for x

3. The Quadratic Formula

The quadratic formula, derived from completing the square:

The discriminant (\( b^2 - 4ac \)) determines the nature of the roots:

• Positive discriminant: Two real roots
• Zero discriminant: One real root (repeated)
• Negative discriminant: Two complex roots

4. Using the Calculator

Instructions: Enter the coefficients a, b, and c from your quadratic equation. The calculator will provide the solutions and show the step-by-step process of completing the square.

5. Frequently Asked Questions (FAQ)

Q1: Why use completing the square instead of factoring?
A: Completing the square works for all quadratic equations, while factoring only works when the equation can be easily factored.

Q2: What if my equation has complex solutions?
A: The calculator will display the complex solutions in the form a ± bi.

Q3: Can I use this for equations where a = 0?
A: No, this is for quadratic equations only (a ≠ 0). If a=0, it becomes a linear equation.

Q4: How accurate are the solutions?
A: Solutions are calculated with floating-point precision and rounded to 4 decimal places.

Q5: Can I see the intermediate steps?
A: Yes, the calculator shows each step of the completing the square process.