Completing the Square

Summary

Completing the square rewrites a quadratic expression as a perfect square plus a constant. In analytic geometry it reveals centers and radii of circles and the standard forms of other conics.

Prerequisites

Polynomial algebra (expanding and factoring). Helpful: Polynomials and Rational Functions. Hub: Analytic Geometry.

Object / Concept

Given a monic quadratic in one variable,

x^{2} + b x + c,

completing the square produces

x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c .

For a leading coefficient $a \neq 0$ ,

a x^{2} + b x + c = a [(x + \frac{b}{2 a})^{2} - (\frac{b}{2 a})^{2}] + c .

Coordinate System

Usually the Cartesian plane or space, when rewriting equations such as $x^{2} + y^{2} + D x + E y + F = 0$ .

Notation

Symbol	Meaning
$a, b, c$	coefficients of $a x^{2} + b x + c$
$(h, k)$	center after rewriting in completed-square form

Conditions / Assumptions

Real coefficients unless stated otherwise.
For the monic formula, divide through by $a$ first when $a \neq 1$ .
Completing the square does not change the solution set of an equation; both sides must be handled consistently.

Equations

One variable (monic)

x^{2} + b x = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} .

Circle general form

x^{2} + y^{2} + D x + E y + F = 0

becomes, after completing the square in $x$ and in $y$ ,

(x + \frac{D}{2})^{2} + (y + \frac{E}{2})^{2} = \frac{D^{2} + E^{2} - 4 F}{4},

provided the right-hand side is positive (circle), zero (point), or negative (empty over the reals).

Procedure

Group $x$ terms (and $y$ terms if present); move constants to the other side if solving/identifying.
Factor out the leading coefficient of each quadratic variable if it is not $1$ .
Add and subtract $(\frac{b}{2 a})^{2}$ inside each group (or add the same quantity to both sides of an equation).
Write each group as a square and simplify constants.
Read off center, radius, or other geometric data.

Worked Example

Rewrite $x^{2} + y^{2} - 6 x + 4 y - 3 = 0$ and identify the circle.

\begin{aligned} x^{2} - 6 x + y^{2} + 4 y & = 3, \\ (x^{2} - 6 x + 9) + (y^{2} + 4 y + 4) & = 3 + 9 + 4, \\ (x - 3)^{2} + (y + 2)^{2} & = 16. \end{aligned}

Center $(3, - 2)$ , radius $4$ .

Common Mistakes

Forgetting to multiply the completed-square constant by $a$ when $a \neq 1$ .
Adding $(\frac{b}{2})^{2}$ to only one side of an equation.
Sign errors: $x^{2} - 6 x$ completes with $(x - 3)^{2}$ , not $(x + 3)^{2}$ .
Treating a nonpositive right-hand side as a circle of real radius without checking.

Connections

Related: Distance and Midpoint, Conic Sections, Polynomials and Rational Functions
Next: Conic Sections, Lines and Planes
Later: quadratic forms in Linear Algebra

References

Completing the square for conics and quadratics is standard in OpenStax Precalculus analytic geometry material.^[1]

OpenStax, Precalculus 2e, Analytic Geometry / Quadratic Functions, https://openstax.org/details/books/precalculus-2e ↩︎