Completing the Square Calculator

Rewrite a quadratic ax² + bx + c in vertex form a(x − h)² + k by completing the square, and read off the vertex.

x2 + x +
Vertex form —
h —
k —

How to Complete the Square

Enter the coefficients a, b and c.

1

Enter the coefficients

Type a, b and c from ax² + bx + c. a cannot be 0.

2

Read the vertex form

The quadratic is rewritten as a(x − h)² + k.

3

Read the vertex

The vertex of the parabola is the point (h, k).

Frequently Asked Questions

What does completing the square do?
Completing the square rewrites a quadratic ax² + bx + c as a(x − h)² + k, a form that makes the vertex and the graph easy to read off. The x² and x terms are combined into a perfect square (x − h)², and the leftover constant becomes k. This is the same idea behind deriving the quadratic formula, and it works even when the quadratic doesn’t factor over the integers.
How do you find h and k?
The vertex x-coordinate is h = −b ⁄ (2a), the point where the parabola turns. Substituting that value back into ax² + bx + c gives k = c − b² ⁄ (4a), the quadratic’s value at the vertex. The calculator computes both directly from a, b and c, so you get vertex form without doing the algebra by hand.
What is vertex form used for?
Vertex form shows the parabola’s turning point directly, which helps with graphing, finding maxima or minima, and solving the equation. Because a(x − h)² + k separates the shift (h, k) from the scaling factor a, you can sketch the curve by moving the basic parabola y = x² to the vertex and stretching it accordingly. It’s also the quickest way to answer optimization questions, such as the maximum height of a projectile or the minimum cost of a business model, since the answer is exactly the vertex.
Is my data sent anywhere?
No — the completing-the-square calculator runs entirely in your browser using JavaScript, with no server round-trip. Your coefficients are never uploaded, logged or stored. The tool keeps working even without an internet connection.

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