About this tool
The Completing the Square Calculator converts a quadratic ax² + bx + c into vertex form a(x − h)² + k. It reads the coefficients a, b, and c from the expression you enter (a must be non-zero), then computes the vertex directly: h = −b/(2a) and k = c − b²/(4a), which is equivalent to k = c − a·h². The vertex form is assembled with correct signs, so a positive h prints as (x − h) and a negative h prints as (x + |h|).
The canonical completing-the-square steps are shown as well: factor a from the x-terms to get a(x² + (b/a)x) + c, then add and subtract the square of half the linear coefficient, (b/(2a))², inside the parentheses to form the perfect square. Alongside the vertex form the tool reports the vertex (h, k), the axis of symmetry x = h, and whether the parabola opens up (a > 0) or down (a < 0).
All coefficients are kept as exact fractions with gcd reduction, so vertices at rational points stay exact, and everything runs locally in your browser with an eval-free parser. Enter a quadratic such as 'x^2 - 6x + 5' and read the vertex form, vertex, axis of symmetry, and the full step-by-step derivation.