What Is Completing the Square?
Completing the square is an algebraic technique used to convert a quadratic expression of the form x² + bx + c into a perfect square trinomial (x + d)² + e.
The 4 Universal Steps
- Ensure the coefficient of x² is 1 (if not, divide the entire equation by a).
- Move the constant term c to the right side of the equation.
- Take half of the coefficient of x (b/2), square it, and add (b/2)² to BOTH sides.
- Factor the left side as a perfect square binomial (x + b/2)² and solve by taking square roots.
Example Problem Walkthrough
Solve: x² + 6x − 7 = 0
- Move constant: x² + 6x = 7
- Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7 + 9 → (x + 3)² = 16
- Take square roots: x + 3 = ±4 → x = 1 or x = -7.