1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by converting them into perfect square trinomial form. This technique is fundamental in algebra and provides insights into the properties of quadratic functions.

2. How Does the Calculator Work?

The calculator uses the quadratic formula and completing the square method:

Where:

• \( a, b, c \) — Coefficients of the quadratic equation
• \( h = -\frac{b}{2a} \) — Horizontal shift
• \( k = c - ah^2 \) — Vertical shift

Explanation: The process transforms the quadratic equation into vertex form, revealing the parabola's vertex and axis of symmetry.

3. Importance of Completing the Square

Details: This method is essential for deriving the quadratic formula, graphing parabolas, solving optimization problems, and in calculus for integration techniques.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The calculator will provide both the roots and the completed square form.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant is negative?
A: The calculator will show complex roots in the form a ± bi, indicating the parabola doesn't intersect the x-axis.

Q2: Why is completing the square useful?
A: It reveals the vertex of the parabola directly, making it easier to graph and analyze the quadratic function.

Q3: What happens if a = 0?
A: The equation becomes linear, not quadratic. The calculator requires a ≠ 0.

Q4: Can this method solve all quadratic equations?
A: Yes, all quadratic equations can be solved by completing the square, though factoring or the quadratic formula may be more efficient in some cases.

Q5: How is this related to the vertex form?
A: The completed square form a(x-h)² + k is identical to the vertex form, where (h,k) is the parabola's vertex.