1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by rewriting them in vertex form. This technique is fundamental in algebra and provides insights into the properties of quadratic functions, including finding their maximum or minimum points.

2. How Does the Calculator Work?

The calculator uses the standard quadratic formula:

Where:

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

Explanation: The calculator first finds solutions using the quadratic formula, then shows the step-by-step process of completing the square to convert the equation to vertex form.

3. Importance of Completing the Square

Details: Completing the square is essential for deriving the quadratic formula, graphing parabolas, solving quadratic equations, and in calculus for integration. It reveals the vertex of the parabola directly from the equation.

4. Using the Calculator

Tips: Enter coefficients a, b, and c of your quadratic equation (ax² + bx + c = 0). Coefficient a cannot be zero. The calculator will show both the solutions and the completed square form.

5. Frequently Asked Questions (FAQ)

Q1: What if my equation has complex solutions?
A: The calculator will display solutions in complex form (with 'i' for imaginary unit) when the discriminant is negative.

Q2: Why is the coefficient a important?
A: If a=0, the equation becomes linear, not quadratic. The method of completing the square only works for quadratic equations.

Q3: What does the vertex form tell us?
A: The vertex form a(x-h)² + k directly shows the vertex of the parabola at point (h, k) and whether it opens upward or downward.

Q4: Can I use this for any quadratic equation?
A: Yes, this method works for all quadratic equations, though it's most useful when you need the vertex information.

Q5: How is this related to the quadratic formula?
A: The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0.