Completing the Square Calculator -- Vertex Form

Complete The Square

Enter the coefficients of your quadratic expression ax² + bx + c to convert it to vertex form a(x - h)² + k with detailed step-by-step work.

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What Is Completing The Square?

Completing the square is an algebraic technique for rewriting a quadratic expression from standard form ax² + bx + c into vertex form a(x - h)² + k.

This transformation reveals the vertex of the parabola at point (h, k), which is the parabola's minimum (if a > 0) or maximum (if a < 0) value. Vertex form makes graphing easier and is essential for understanding the geometry of quadratic functions.

The Process

The goal is to create a perfect square trinomial from the x² and x terms. Here are the steps:

1. Factor out the leading coefficient a from the x² and x terms (but not the constant c)
2. Take half of the new b coefficient (after factoring out a), then square it
3. Add and subtract this value inside the factored expression
4. Factor the perfect square trinomial into (x - h)² form
5. Simplify by distributing a and combining constants to get a(x - h)² + k

Why It Matters

• Reveals the vertex — The point (h, k) is immediately visible in vertex form
• Makes graphing easier — Vertex form shows transformations from the basic parabola y = x²
• Derives the quadratic formula — Completing the square on ax² + bx + c = 0 yields the quadratic formula
• Solves optimization problems — Finding maximum or minimum values becomes straightforward
• Analyzes motion — In physics, projectile motion equations often benefit from vertex form

Example Walkthrough

Let's complete the square for x² + 6x + 5:

1. Since a = 1, we don't need to factor anything out
2. Take half of b: 6/2 = 3, then square it: 3² = 9
3. Add and subtract 9: x² + 6x + 9 - 9 + 5
4. Factor the perfect square: (x + 3)² - 9 + 5
5. Simplify: (x + 3)² - 4

The vertex form is (x + 3)² - 4, which means the vertex is at (-3, -4). Notice that the sign of h is opposite to what appears in the parentheses.

When a ≠ 1

For 2x² + 12x + 10:

1. Factor out a = 2 from the x² and x terms: 2(x² + 6x) + 10
2. Take half of 6: 6/2 = 3, square it: 3² = 9
3. Add and subtract 9 inside: 2(x² + 6x + 9 - 9) + 10
4. Factor: 2((x + 3)² - 9) + 10
5. Distribute: 2(x + 3)² - 18 + 10 = 2(x + 3)² - 8

The vertex is at (-3, -8).

Vertex Form vs Standard Form

Common Mistakes

• Forgetting to factor out a first — When a ≠ 1, you must factor it out from the x² and x terms before completing the square on the simpler expression inside.
• Sign confusion with h — In vertex form a(x - h)² + k, the vertex x-coordinate is h, but the sign in the parentheses is opposite: (x - 3) means h = 3, (x + 3) means h = -3.
• Distributing a incorrectly — When you add/subtract (b/2a)² inside the factored expression, remember that distributing a back through means you're actually adding/subtracting a×(b/2a)² = b²/4a outside.
• Losing the constant term — Keep track of c throughout the process. It gets combined with the adjustment term at the end.
• Not simplifying fully — Always combine all constants outside the squared term to get the cleanest vertex form.

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Completing The Square Calculator FAQ