Complete the Square Calculator

Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Solutions:

Completed Square Form:

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1. What is Completing the Square?

Completing the square is an algebraic technique used to solve quadratic equations, rewrite quadratic expressions, and analyze properties of quadratic functions. It transforms a quadratic equation from standard form (ax² + bx + c) to vertex form (a(x - h)² + k).

2. How Does the Calculator Work?

The calculator uses the following process:

\[ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \]

Where:

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

Explanation: The method involves creating a perfect square trinomial from the quadratic expression, which can then be written as a squared binomial.

3. Importance of Completing the Square

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

5. Frequently Asked Questions (FAQ)

Q1: When should I use completing the square instead of the quadratic formula?
A: Completing the square is useful when you need the vertex form of the equation (for graphing) or when deriving the quadratic formula itself. The quadratic formula is generally faster for just finding roots.

Q2: What if my 'a' coefficient is not 1?
A: The calculator handles any non-zero value of a. The process involves factoring out a from the x² and x terms before completing the square.

Q3: What do complex solutions mean?
A: When the discriminant (b² - 4ac) is negative, the quadratic equation has no real solutions but two complex conjugate solutions.

Q4: How is the vertex related to the completed square form?
A: In the form a(x - h)² + k, the vertex of the parabola is at (h, k).

Q5: Can this method be used for higher degree polynomials?
A: Completing the square is specific to quadratic (degree 2) polynomials. Other techniques are needed for higher degree equations.