| Version current |
Version 10 |
| 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*} |
\begin{eqnarray*} |
| x^2+xy &=& x^2+xy+ 0\\ |
x^2+xy &=& x^2+xy+ 0\\ |
| &=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\ |
&=& x^2+xy+ \frac{y^2}{4}-\frac{y^2}{4}\\ |
| &=& \left(x+\frac{y}{2}\right)^2-\frac{y^2}{4}. |
&=& \left(x+\frac{y}{2}\right)^2-\frac{y^2}{4}. |
| \end{eqnarray*} |
\end{eqnarray*} |
| This manipulation is called \emph{completing the square} \cite{adams} in |
This manipulation is called \emph{completing the square} \cite{adams} in |
| $x^2+xy$, or completing the square $x^2$. |
$x^2+xy$, or completing the square $x^2$. |
| |
|
| Replacing $y$ by $-y$, we also have |
Replacing $y$ by $-y$, we also have |
| $$x^2-xy = \left(x-\frac{y}{2}\right)^2-\frac{y^2}{4}.$$ |
$$x^2-xy = \left(x-\frac{y}{2}\right)^2-\frac{y^2}{4}.$$ |
|
|
| Here are some applications of this method: |
Here are some applications of this method: |
| \begin{itemize} |
\begin{itemize} |
| \item |
\item |
| \PMlinkname{Derivation of the solution formula to the quadratic equation}{DerivationOfQuadraticFormula}. |
\PMlinkname{Derivation of the solution formula to the quadratic equation}{DerivationOfQuadraticFormula}. |
| \item Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle |
\item Putting the general equation of a circle, ellipse, or hyperbola into standard form, e.g. the circle |
| \begin{align*} |
\begin{align*} |
| x^2+y^2+2x+4y=5\Rightarrow (x+1)^2 + (y+2)^2= 10, |
x^2+y^2+2x+4y=5\Rightarrow (x+1)^2 + (y+2)^2= 10, |
| \end{align*} |
\end{align*} |
| from which it is frequently easier to read off important information (the center, radius, etc.) |
from which it is frequently easier to read off important information (the center, radius, etc.) |
| \item Completing the square can also be used to find the extremal value |
\item Completing the square can also be used to find the extremal value |
| of a quadratic polynomial \cite{thompson} without calculus. |
of a quadratic polynomial \cite{thompson} without calculus. |
| Let us illustrate this for the polynomial $p(x)=4x^2+8x+9$. |
Let us illustrate this for the polynomial $p(x)=4x^2+8x+9$. |
| Completing the square yields |
Completing the square yields |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| p(x) &=& (2x+2)^2-4 +9 \\ |
p(x) &=& (2x+2)^2-4 +9 \\ |
| &=& (2x+2)^2+5 \\ |
&=& (2x+2)^2+5 \\ |
| &\ge & 5, |
&\ge & 5, |
| \end{eqnarray*} |
\end{eqnarray*} |
| since $(2x+2)^2\ge 0$. Here, equality holds if and |
since $(2x+2)^2\ge 0$. Here, equality holds if and |
| only if $x=-1$. |
only if $x=-1$. |
| Thus $p(x)\ge 5$ for all $x\in \sR$, and $p(x)=5$ if and only if |
Thus $p(x)\ge 5$ for all $x\in \sR$, and $p(x)=5$ if and only if |
| $x=-1$. |
$x=-1$. |
| It follows that $p(x)$ has a global minimum at $x=-1$, where $p(-1)=5$. |
It follows that $p(x)$ has a global minimum at $x=-1$, where $p(-1)=5$. |
| \item Completing the square can also be used as an integration technique |
\item Completing the square can also be used as an integration technique |
| to integrate, for example the function $\displaystyle \frac{1}{4x^2+8x+9}$ \cite{adams}. |
to integrate, for example the function $\displaystyle \frac{1}{4x^2+8x+9}$ \cite{adams}. |
| \end{itemize} |
\end{itemize} |
|
|
|
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem {adams} R. Adams, \emph{Calculus, a complete course}, |
\bibitem {adams} R. Adams, \emph{Calculus, a complete course}, |
| Addison-Wesley Publishers Ltd, 3rd ed. |
Addison-Wesley Publishers Ltd, 3rd ed. |
| \bibitem {thompson} |
\bibitem {thompson} |
|
\emph{Matematiklexikon} (in Swedish),
|
\emph{Matematik Lexikon} (in Swedish),
|
| J. Thompson, T. Martinsson, Wahlstr\"om \& Widstrand, 1991. |
J. Thompson, T. Martinsson, Wahlstr\"om \& Widstrand, 1991. |
| \end{thebibliography} |
\end{thebibliography} |
|
|
| (Anyone has an English reference?) |
(Anyone has an English reference?) |
