Completing the square converts a quadratic from standard form (ax² + bx + c) into vertex form (a(x − h)² + k), instantly revealing the parabola's vertex without graphing or using the quadratic formula. It is also the technique used to derive the quadratic formula itself.

How completing the square works

Start with ax² + bx + c. If a = 1, group the x² and x terms, take half of the x-coefficient, and square it — that square is exactly what's needed to turn the group into a perfect-square trinomial. Add and subtract it so the expression's value doesn't change, then rewrite the perfect square as (x − h)² and combine the leftover constants into k.

If a ≠ 1, factor a out of the first two terms before completing the square inside the parentheses, then distribute a back out over the constant that gets subtracted. The formulas h = −b/(2a) and k = c − b²/(4a) give the same result directly, without writing out every algebra step by hand.

Inputs and what they mean

Enter a, b, and c exactly as they appear in your quadratic ax² + bx + c. a is the leading coefficient and cannot be 0 — a value of 0 would make the expression linear, with no square to complete. b is the coefficient of x, and c is the constant term. All three can be positive, negative, decimal, or whole numbers.

The two outputs, h and k, together form the vertex (h, k). h tells you the parabola's axis of symmetry (x = h), and k tells you the minimum or maximum value the quadratic can take.

Limits and edge cases

This calculator handles real-number coefficients for a standard quadratic; it does not solve for complex roots or handle quadratics in more than one variable. When a is very close to 0 (but not exactly 0), h and k can become extremely large, since both formulas divide by a — double-check your coefficients if the result looks unexpectedly huge.

Completing the square is closely related to solving quadratic equations: setting the vertex form equal to 0 and solving for x is exactly how the quadratic formula itself is derived. If you need the roots of ax² + bx + c = 0 rather than the vertex form, use the Quadratic Formula Calculator instead.