Completing the Square Calculator with Steps

How it works

ax² + bx + c = 0 → x² + (b/a)x = −c/a → add (b/2a)² → (x + b/2a)² = (b² − 4ac)/4a² → x = −b/2a ± √(b² − 4ac)/2a · vertex form a(x − h)² + k with h = −b/2a, k = c − b²/4a

Completing the square rewrites a quadratic so that the x terms form a perfect square trinomial. After dividing by a and moving the constant across, you add the square of half the x coefficient to both sides; the left side then factors as (x + b/2a)², and taking square roots finishes the solve. The same algebra gives the vertex form y = a(x − h)² + k, which shows the parabola’s vertex (h, k) directly, and it is exactly how the quadratic formula is derived. All arithmetic here is exact, with fractions and simplified radicals.

Worked example

Solve x² + 6x + 5 = 0. Move the constant: x² + 6x = −5. Half of 6 is 3 and 3² = 9; add it to both sides: x² + 6x + 9 = 4, so (x + 3)² = 4. Square roots: x + 3 = ±2, giving x = −1, x = −5. The vertex form is y = (x + 3)² − 4 with vertex (−3, −4).

Sources: Completing the square — Wikipedia · Completing the square — Math is Fun

Frequently asked questions