Completing the square without guessing
Add the exact term that turns a quadratic expression into a perfect-square trinomial, while preserving the equation's value.
LaTeX article Updated July 13, 2026
The middle term determines the square
Expanding (x + p)² gives x² + 2px + p². Matching 2p with b forces p = b/2, so the constant must be (b/2)².
This is a structural identity, not a factoring guess. It works for every quadratic after the leading coefficient has been normalized to one.
Equations must stay balanced
Adding the completing term changes an expression. In an equation, add the same amount to both sides so the solution set remains unchanged.
If the coefficient of x² is not one, divide the equation by that coefficient or factor it from the quadratic terms before using the half-and-square step.
Completed square form exposes useful features
The form (x − h)² = k makes square-root solving immediate and reveals the vertex of a parabola. It also explains the structure behind the quadratic formula.
After taking square roots, include both signs. Squaring hides whether the original quantity was positive or negative.
Worked example
Common mistakes
- Using b² instead of (b/2)².
- Adding the completing term to only one side.
- Keeping only the positive square root.
Keep these ideas
- Half the linear coefficient and square it.
- Normalize the leading coefficient first.
- Square-root solving needs ±.