1. What is Completing the Square?

Completing the square is a technique for converting a quadratic equation from standard form (ax² + bx + c) to vertex form (a(x - h)² + k). This method is useful for solving quadratic equations, graphing parabolas, and deriving the quadratic formula.

2. How Does the Calculator Work?

The calculator uses the completing the square formula:

Where:

• \( a \) — Coefficient of x²
• \( b \) — Coefficient of x
• \( c \) — Constant term

Explanation: The process involves creating a perfect square trinomial from the quadratic and linear terms, then adjusting the constant term to maintain equality.

3. Importance of Completing the Square

Details: Completing the square is essential for finding the vertex of a parabola, solving quadratic equations, and is the basis for the quadratic formula. It's also used in calculus for integration and in deriving equations of circles.

4. Using the Calculator

Tips: Enter the coefficients of your quadratic equation (a, b, and c). The calculator will return the equation in vertex form. Note that a cannot be zero.

5. Frequently Asked Questions (FAQ)

Q1: Why complete the square instead of using the quadratic formula?
A: Completing the square gives the vertex form directly, which is useful for graphing. The quadratic formula is better for just finding roots.

Q2: What if my 'a' coefficient is not 1?
A: The calculator handles any non-zero 'a' value by factoring it out first before completing the square.

Q3: Can I use this for complex numbers?
A: The calculator works with real numbers. For complex numbers, the process is similar but the interpretation differs.

Q4: How is this related to the vertex of a parabola?
A: The vertex form a(x - h)² + k shows the vertex at (h, k).

Q5: What's the connection to the quadratic formula?
A: The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0.