Completing the Square Calculator — Vertex Form (Free)

Use this free completing the square calculator to rewrite a quadratic in vertex form. Enter a, b and c and you’ll get a(x − h)² + k, the vertex (h, k), the axis of symmetry, and the real roots — with each step of completing the square shown.

How to use the calculator

1. Enter the coefficients a, b and c of ax² + bx + c.
2. Press Calculate.
3. Read the vertex form, the vertex, and the roots, with the steps.

Completing the square

Factor a from the x-terms, take half of the resulting x-coefficient and square it, then add and subtract it to form a perfect square. The result is vertex form, where h = −b/(2a) and k = c − b²/(4a).

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Frequently asked questions

What is vertex form?

a(x − h)² + k, where (h, k) is the vertex of the parabola. It makes the vertex and axis of symmetry easy to read off.

How do you complete the square?

Take half the coefficient of x, square it, and add and subtract that value so part of the expression becomes a perfect square trinomial.

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Does it also give the roots?

Yes — from vertex form it solves x = h ± √(−k/a), or reports when there are no real roots.

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Read the full lesson: learn the method step by step.

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