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Completing the Square Calculator

Convert quadratic equations from standard form (ax^2 + bx + c) to vertex form a(x + h)^2 + k using the completing the square method. Shows every step of the process.

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Formula

ax^2 + bx + c = a(x + b/2a)^2 + (c - b^2/4a)

Completing the square rewrites a quadratic in vertex form by creating a perfect square trinomial. Half the coefficient of x, square it, add and subtract inside the expression. The vertex (h, k) reveals the minimum or maximum point and axis of symmetry.

Worked Example

Complete the square for x^2 + 6x + 5: Step 1: Half of 6 = 3 Step 2: 3^2 = 9 Step 3: x^2 + 6x + 9 - 9 + 5 = (x + 3)^2 - 4 Vertex form: (x + 3)^2 - 4 Vertex: (-3, -4) Minimum value: -4

What Is Completing the Square?

Completing the square is a powerful algebraic technique used to rewrite a quadratic expression from its standard form, ax^2 + bx + c, into vertex form, a(x + h)^2 + k. The core idea is to manipulate the expression to create a perfect square trinomial, which is a trinomial that can be factored as (x + d)^2. This method involves taking half of the coefficient of the x term, squaring it, and then adding and subtracting this value to the expression to maintain its equality. By strategically adding and subtracting the same value, you can group terms to form the perfect square while isolating a constant that becomes the 'k' in the vertex form. This transformation is crucial for easily identifying the vertex, axis of symmetry, and direction of opening of a parabola, making it an invaluable tool for graphing and analyzing quadratic functions.
  • Transforms quadratic equations into vertex form, a(x + h)^2 + k.
  • Directly reveals the vertex (h, k), which is the parabola's minimum or maximum point.
  • Identifies the axis of symmetry, x = h, providing insights into the parabola's structure.
  • Can be used to solve quadratic equations by isolating x after forming the perfect square.
  • A fundamental skill for advanced algebra, pre-calculus, and calculus studies.

Understanding completing the square provides a deeper insight into quadratic functions and their graphical representations. Use the Calculory.AI tool to practice converting equations and solidify your comprehension of every step involved.

You can also calculate changes using our Polynomial Factoring Calculator or Partial Derivative Calculator.

Frequently Asked Questions

Why complete the square instead of using the quadratic formula?

Completing the square gives you the vertex form, which directly reveals the vertex (minimum or maximum point), axis of symmetry, and direction of opening. The quadratic formula only gives roots.

What is vertex form?

Vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex. If a > 0, the parabola opens upward and k is the minimum. If a < 0, it opens downward and k is the maximum.

When is completing the square useful?

It is essential for deriving the quadratic formula, converting conic section equations to standard form, solving optimization problems, and understanding the graph of a quadratic.

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