Evaluate $(2009)^2 - (2008)(2010)$.

$$\small{\text{Evaluate $(2009)^2 - (2008)(2010). $}}$$

I.

$$\small{\text{$
\begin{array}{rcl}
(2009)^2 - (2008)(2010) \\
&=& 2009^2-(2009-1)(2009+1)\\
&=& 2009^2-(2009^2-1)\\
&=& 2009^2 - 2009^2 + 1 \\
&=& 1
\end{array}
$}}$$

II.

$$\small{\text{$
\begin{array}{rcl}
(2009)^2 - (2008)(2010) \\
&=& (2008+1)^2 - 2008\cdot 2010 \\
&=& 2008^2 + 2\cdot 2008 + 1 - 2008\cdot 2010\\
&=& 2008^2 - 2008 (-2+2010) + 1 \\
&=& 2008^2 - 2008^2 + 1 \\
&=& 1
\end{array}
$}}$$

III.

$$\small{\text{$
\begin{array}{rcl}
(2009)^2 - (2008)(2010) \\
&=& (2010-1)^2 - 2008\cdot 2010 \\
&=& 2010^2 - 2\cdot 2010 + 1 - 2008\cdot 2010\\
&=& 2010^2 - 2010 (2+2008) + 1 \\
&=& 2010^2 - 2010^2 + 1 \\
&=& 1
\end{array}
$}}$$

 

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