1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by rewriting them in perfect square form. It's a fundamental technique in algebra that's used to derive the quadratic formula and understand the properties of quadratic functions.

2. How Does the Calculator Work?

The calculator uses the completing the square formula:

Where:

• \( a \) — Coefficient of x² (must not be zero)
• \( b \) — Coefficient of x
• \( c \) — Constant term

Explanation: The formula transforms the quadratic equation ax² + bx + c = 0 into vertex form, revealing the roots (solutions) of the equation.

3. Importance of Completing the Square

Details: This method is crucial for understanding quadratic functions, finding their maximum/minimum points (vertex), and solving equations that can't be factored easily. It's also the basis for the quadratic formula.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0. The calculator will provide real or complex solutions.

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex solutions?
A: Complex solutions (with 'i') occur when the discriminant is negative, meaning the parabola doesn't cross the x-axis.

Q2: Why can't coefficient a be zero?
A: If a=0, the equation becomes linear (bx + c = 0), not quadratic, and requires a different solving method.

Q3: How is this related to the quadratic formula?
A: The quadratic formula is derived by completing the square for the general quadratic equation.

Q4: When should I use completing the square vs factoring?
A: Use completing the square when the quadratic doesn't factor easily, or when you need the vertex form.

Q5: Can this method be used for graphing?
A: Yes, the completed square form directly reveals the vertex of the parabola, making it ideal for graphing.