Completing the Square Calculator

Reviewed by CalcMulti Editorial Team·Last updated: March 2026·← Algebra Hub

Completing the square converts ax² + bx + c into vertex form a(x − h)² + k, immediately revealing the vertex (h, k) and axis of symmetry. It also yields the roots by solving a(x−h)² = −k.

This method is foundational in algebra: it derives the quadratic formula, solves quadratic equations without memorising the formula, and rewrites parabola equations for graphing. Enter your coefficients below for a full step-by-step solution.

The step-by-step process for completing the square with x² + bx + c (when a = 1): Step 1 — move the constant to the right side: x² + bx = −c. Step 2 — add (b/2)² to both sides: x² + bx + (b/2)² = −c + (b/2)². Step 3 — the left side is now a perfect square: (x + b/2)² = −c + (b/2)². Step 4 — take the square root of both sides and solve for x. When a ≠ 1, divide the entire equation by a first, then proceed with steps 1–4.

Worked example — complete the square for 2x² − 8x + 3: First divide by 2: x² − 4x + 3/2. Move constant: x² − 4x = −3/2. Add (−4/2)² = 4 to both sides: x² − 4x + 4 = −3/2 + 4 = 5/2. Write as square: (x − 2)² = 5/2. So the vertex form is 2(x − 2)² − 5/2, with vertex at (2, −5/2). The roots are x = 2 ± √(5/2) = 2 ± √(10)/2.

Completing the square has three main uses: (1) Finding the vertex of a parabola for graphing — the vertex (h, k) gives the minimum or maximum value. (2) Solving quadratic equations when the discriminant is positive — produces exact roots without the quadratic formula. (3) Rewriting conic section equations — circles, ellipses, and hyperbolas are all written in standard form by completing the square in both x and y.