1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by rewriting them in perfect square 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 completing the square formula:

Where:

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

Explanation: The method transforms the quadratic equation ax² + bx + c = 0 into the form (x + d)² = e, where d and e are constants derived from a, b, and c.

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 coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0. The calculator will provide both solutions (if real) and the completed square form.

5. Frequently Asked Questions (FAQ)

Q1: What if I get a negative under the square root?
A: This means there are no real solutions (the parabola doesn't cross the x-axis). The solutions would be complex numbers.

Q2: Can a be zero?
A: No, if a is zero, the equation is linear, not quadratic. The calculator requires a non-zero value for a.

Q3: Why is completing the square useful?
A: It's used to derive the quadratic formula, find vertex form of a parabola, and solve certain integrals in calculus.

Q4: What's the advantage over the quadratic formula?
A: Completing the square gives the vertex form directly, which is useful for graphing and optimization problems.

Q5: How is this related to vertex form?
A: The completed square form a(x - h)² + k directly shows the vertex (h, k) of the parabola.