1. What is Completing the Square?

Completing the square is a method for solving quadratic equations by converting them into perfect square form. This technique is fundamental in algebra and is used to derive the quadratic formula.

2. How Does the Calculator Work?

The calculator uses the quadratic formula and shows the steps for completing the square:

Where:

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

Explanation: The method transforms the equation ax² + bx + c = 0 into the form (x + d)² = e, making it easier to solve.

3. Importance of Completing the Square

Details: This method is essential for solving quadratic equations, graphing parabolas, and understanding the derivation of the quadratic formula. It's also used in calculus for integration.

4. Using the Calculator

Tips: Enter the coefficients a, b, and c from your quadratic equation (ax² + bx + c = 0). The calculator will show both the solutions and the step-by-step process.

5. Frequently Asked Questions (FAQ)

Q1: What if a = 0?
A: If a = 0, the equation is linear (not quadratic) and can be solved as bx + c = 0.

Q2: What do complex solutions mean?
A: Complex solutions occur when the discriminant (b²-4ac) is negative, indicating the parabola doesn't cross the x-axis.

Q3: Why complete the square instead of factoring?
A: Completing the square works for all quadratic equations, while factoring only works when the equation can be easily factored.

Q4: Can this method be used for any quadratic?
A: Yes, completing the square can solve any quadratic equation, though it may be more complex than other methods in some cases.

Q5: How is this related to vertex form?
A: Completing the square converts standard form (ax²+bx+c) to vertex form a(x-h)²+k, revealing the vertex (h,k) of the parabola.