1. What is Completing The Square?

Completing the square is a method for solving quadratic equations by rewriting them in a perfect square form. It's a fundamental technique in algebra that provides an alternative to the quadratic formula.

2. How Does the Calculator Work?

The calculator uses the completing the square formula:

Where:

• \( a \) — Coefficient of x² term (must not be zero)
• \( b \) — 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 technique is essential for deriving the quadratic formula, graphing parabolas, solving optimization problems, and in calculus for integration of rational functions.

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 (b² - 4ac) is negative, meaning no real solutions exist.

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

Q3: How is this different from the quadratic formula?
A: Both methods give the same solutions. The quadratic formula is derived by completing the square on the general quadratic equation.

Q4: When is completing the square preferred?
A: It's useful when you need the vertex form of a quadratic (for graphing) or when dealing with certain integrals in calculus.

Q5: Can this 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.