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
  1. Halve the x-coefficient, then square it: (-3)² = 9
  2. Add and subtract that square inside: (x − 3)²
  3. Group as a perfect square and simplify the constant: (x − 3)² − 4
Parabola in vertex formA parabola plotted from the quadratic, with its vertex, axis of symmetry, and minimum or maximum value highlighted. Completing the square rewrites the equation so the vertex can be read off directly. (x − 3)² − 4.x = 315(3, -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