Completing the square

Math

Definition

Completing the square is a method for rewriting a quadratic expression in the form (x + p)² + q, which makes it easy to solve quadratic equations or convert to vertex form. It works by adding and subtracting a specific constant to create a perfect square trinomial.

How It Works

  1. Start with ax² + bx + c; if a ≠ 1, factor out a from the first two terms
  2. Take half of the coefficient of x, then square it
  3. Add and subtract that value inside the expression
  4. Factor the perfect square trinomial
  5. Simplify the remaining constant

Examples

  • x² + 6x + 5 = 0 becomes (x + 3)² - 4 = 0, so x = -3 ± 2
  • Converting y = x² - 4x + 7 to vertex form y = (x - 2)² + 3
  • Deriving the quadratic formula starts with completing the square on ax² + bx + c = 0
Key Fact

For x² + bx, add (b/2)² to complete the square.

Related Terms

Quadratic formula Vertex form Solving by factoring Discriminant

Study This Concept

Practice completing the square with free review games in these units:

Algebra 1 Unit 8: Quadratic Equations Algebra 2 Unit 3: Quadratic Functions