completing the square

Let us consider the expression x2+xy, where x and y are real (or complex) numbers. Using the formulaMathworldPlanetmathPlanetmath


we can write

x2+xy = x2+xy+0
= x2+xy+y24-y24
= (x+y2)2-y24.

This manipulation is called completing the square [1] in x2+xy, or completing the square x2.

Replacing y by -y, we also have


Here are some applications of this method:

  • of the solution formula to the quadratic equation.

  • Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle


    from which it is frequently easier to read off important information (the center, radius, etc.)

  • Completing the square can also be used to find the extremal value of a quadratic polynomial [2] without calculus. Let us illustrate this for the polynomialPlanetmathPlanetmath p(x)=4x2+8x+9. Completing the square yields

    p(x) = (2x+2)2-4+9
    = (2x+2)2+5

    since (2x+2)20. Here, equality holds if and only if x=-1. Thus p(x)5 for all x, and p(x)=5 if and only if x=-1. It follows that p(x) has a global minimumMathworldPlanetmath at x=-1, where p(-1)=5.

  • Completing the square can also be used as an integration technique to integrate, for example the function 14x2+8x+9 [1].


  • 1 R. Adams, Calculus, a completePlanetmathPlanetmathPlanetmathPlanetmath course, Addison-Wesley Publishers Ltd, 3rd ed.
  • 2 Matematiklexikon (in Swedish), J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.

(Anyone has an English reference?)

Title completing the square
Canonical name CompletingTheSquare
Date of creation 2013-03-22 13:36:27
Last modified on 2013-03-22 13:36:27
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Algorithm
Classification msc 00A20
Related topic SquareOfSum