Let us consider the expression $x^2+xy$ , where $x$ and $y$ are real (or complex) numbers. Using the formula $$(x+y)^2 = x^2+2xy +y^2$$ we can write \begin{eqnarray*} x^2+xy &=& x^2+xy+ 0\\ &=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\ &=& \left(x+\frac{y}{2}\right)^2-\frac{y^2}{4}. \end{eqnarray*}This manipulation is called completing the square [1] in $x^2+xy$ , or completing the square $x^2$ .

Replacing $y$ by $-y$ , we also have $$x^2-xy = \left(x-\frac{y}{2}\right)^2-\frac{y^2}{4}.$$

Here are some applications of this method:

• Derivation 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 polynomial $p(x)=4x^2+8x+9$ . Completing the square yields \begin{eqnarray*} p(x) &=& (2x+2)^2-4 +9 \\ &=& (2x+2)^2+5 \\ &\ge & 5, \end{eqnarray*}since $(2x+2)^2\ge 0$ . Here, equality holds if and only if $x=-1$ . Thus $p(x)\ge 5$ for all $x\in \sR$ , and $p(x)=5$ if and only if $x=-1$ . It follows that $p(x)$ has a global minimum at $x=-1$ , where $p(-1)=5$ .
• Completing the square can also be used as an integration technique to integrate, for example the function $\displaystyle \frac{1}{4x^2+8x+9}$ [1].

Bibliography

1

R. Adams, Calculus, a complete course, Addison-Wesley Publishers Ltd, 3rd ed.

2

Matematiklexikon (in Swedish), J. Thompson, T. Martinsson, Wahlström & Widstrand, 1991.

(Anyone has an English reference?)