| Version 2 |
Version 1 |
| \newcommand{\sR}[0]{\mathbb{R}} |
Let us consider the expression $x^2+xy$, where |
| Let us consider the expression $x^2+xy$, where |
$x$ and $y$ are real (or complex) numbers. |
| $x$ and $y$ are real (or complex) numbers. |
Using the formula |
| Using the formula |
$$(x+y)^2 = x^2+2xy +y^2$$ |
| $$(x+y)^2 = x^2+2xy +y^2$$ |
we can write |
| we can write |
|
| \begin{eqnarray*} |
|
| x^2+xy &=& x^2+xy+ 0\\ |
|
| &=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\ |
|
| &=& (x+\frac{y}{2})^2-\frac{y^2}{4}. |
|
| \end{eqnarray*} |
|
| This manipulation is called \emph{completing the square} in |
|
| $x^2+xy$, or completing the square $x^2$. |
|
| Replacing $y$ by $-y$, we also have |
|
| $$x^2-xy = (x-\frac{y}{2})^2-\frac{y^2}{4}.$$ |
|
| Here are some uses of this method: |
|
| \begin{itemize} |
|
| \item |
|
| \PMlinkname{Derivation of the solution formula to the quadratic equation}{DerivationOfQuadraticFormula}. |
|
| \item Completing the square can also be used to find the extremal value |
|
| of a quadratic polynomial \cite{thompson}. |
|
| We illustrate this for the polynomial $p(x)=4x^2+8x+9$. Completing the |
|
| square yields |
|
| \begin{eqnarray*} |
\begin{eqnarray*} |
|
p(x) &=& (2x+2)^2-4 +9 \\
|
x^2+xy &=& x^2+xy+ 0\\
|
|
&=& (2x+2)^2+5 \\
|
&=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\
|
| &\ge & 5 |
&=& (x+\frac{y}{2})^2-\frac{y^2}{4}. |
| \end{eqnarray*} |
\end{eqnarray*} |
| since $(2x+2)^2\ge 0$. On the last line, equality is obtained if and |
This manipulation is called \emph{completing the square} in |
| only if $x=-1$. |
$x^2+xy$, or completing the square $x^2$. |
| Thus $p(x)\ge 5$ for all $x\in \sR$, and $p(x)=5$ if and only if |
Replacing $y$ by $-y$, we also have |
| $x=-1$. |
$$x^2-xy = (x-\frac{y}{2})^2+\frac{y^2}{4}.$$ |
| It follows that $p(x)$ has a global minimum at $x=-1$, where $p(-1)=5$. |
As an example, |
| \end{itemize} |
the above method is used to |
| {\bf See also} |
derive |
| \begin{itemize} |
\PMlinkname{the solution formula to the quadratic equation}{DerivationOfQuadraticFormula}. |
| \item E. Weisstein, Eric W. Weisstein's world of mathematics, |
|
| \PMlinkexternal{entry on completing the square}{http://mathworld.wolfram.com/CompletingtheSquare.html}. |
|
| \end{itemize} |
|
| \begin{thebibliography}{9} |
|
| \bibitem {thompson} |
|
| \emph{Matematik Lexikon} (in Swedish), |
|
| ed. J. Thompson, T. Martinsson, Wahlstr\"om \& Widstrand, 1991. |
|
| \end{thebibliography} |
|
| (Anyone has an English reference?) |
|
