                                                                                                                                      
 This is an extension of the things you learned in \emph{Topic 1: Algebra}. We will be using 3 main results and then replacing the    
 $a$'s and $b$'s with \emph{surds}.                                                                                                   
                                                                                                                                      
 \begin{align}                                                                                                                        
 & (a+b)^{2}=a^{2}+2 a b+b^{2} \\                                                                                                     
 & (a-b)^{2}=a^{2}-2 a b+b^{2} \\                                                                                                     
 & (a-b)(a+b)=a^{2}-b^{2}                                                                                                             
 \end{align}                                                                                                                          
                                                                                                                                      
 \begin{examplebox}                                                                                                                   
     \subsection{Examples:}                                                                                                           
     \begin{questions}                                                                                                                
         \Question[5]                                                                                                                 
         \begin{doublespace}                                                                                                          
         \begin{parts}                                                                                                                
         \part \[(\sqrt{5}+\sqrt{3})^{2}=\]                                                                                           
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(\sqrt{5}+\sqrt{3})^{2} = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15}$    
             \end{solutionordottedlines}                                                                                              
         \part \[(3 \sqrt{2}-2 \sqrt{3})^{2}=\]                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(3 \sqrt{2}-2 \sqrt{3})^{2} = (3\sqrt{2})^2 - 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 9\cdot2 - 2\cdot6\sqrt{6} +     
 4\cdot3 = 18 - 12\sqrt{6} + 12$                                                                                                      
             \end{solutionordottedlines}                                                                                              
         \part \[(2 \sqrt{3}+4 \sqrt{6})^{2}=\]                                                                                       
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(2 \sqrt{3}+4 \sqrt{6})^{2} = (2\sqrt{3})^2 + 2(2\sqrt{3})(4\sqrt{6}) + (4\sqrt{6})^2 = 4\cdot3 + 2\cdot8\sqrt{18} +    
 16\cdot6 = 12 + 16\sqrt{18} + 96$                                                                                                    
             \end{solutionordottedlines}                                                                                              
         \part \[(\sqrt{7}-5)(\sqrt{7}+5)=\]                                                                                          
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(\sqrt{7}-5)(\sqrt{7}+5) = (\sqrt{7})^2 - (5)^2 = 7 - 25 = -18$                                                         
             \end{solutionordottedlines}                                                                                              
         \part \[(5 \sqrt{6}-2 \sqrt{5})(5 \sqrt{6}+2 \sqrt{5})=\]                                                                    
             \begin{solutionordottedlines}[2cm]                                                                                       
             $(5 \sqrt{6}-2 \sqrt{5})(5 \sqrt{6}+2 \sqrt{5}) = (5\sqrt{6})^2 - (2\sqrt{5})^2 = 25\cdot6 - 4\cdot5 = 150 - 20 = 130$   
             \end{solutionordottedlines}                                                                                              
         \end{parts}                                                                                                                  
         \end{doublespace}                                                                                                            
     \end{questions}                                                                                                                  
 \end{examplebox}                                                                                                                     
                                                                                                                                      
                                                                                                                                      
