Completing the Square

Rewrite a quadratic in vertex form, step by step

x² − 6x + 5 = (x − 3)² − 4

Vertex form

(x − 3)² − 4

Vertex

(3, -4)

Axis of symmetry

x = 3

Minimum value

-4

Roots

x = 1, 5

Parabola opens upward

Step by step

Halve the x-coefficient, then square it: (-3)² = 9
Add and subtract that square inside: (x − 3)²
Group as a perfect square and simplify the constant: (x − 3)² − 4

About this calculator

The Completing the Square calculator rewrites any quadratic ax² + bx + c into vertex form a(x − h)² + k, showing every step: factoring out a, halving the x-coefficient, squaring it, and grouping the perfect square. It also gives the vertex, axis of symmetry, minimum or maximum value, and real roots, with a parabola visual. Useful for algebra homework, graphing, and deriving the quadratic formula.

Common examples

x² − 6x + 5 → (x − 3)² − 4, vertex (3, −4), roots x = 1 and x = 5
x² + 6x + 11 → (x + 3)² + 2, vertex (−3, 2), no real roots
2x² − 8x + 3 → 2(x − 2)² − 5, vertex (2, −5)
x² + 4x + 4 → (x + 2)², a perfect square with double root x = −2
−x² + 2x + 3 → −(x − 1)² + 4, vertex (1, 4), opens downward