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KCC's Quizzes AQQ253 about a funny equation



		

			

									
					    KCC
					
							

			
				
									
											
			
			
				
  •  Analog Employees  

			
			
			    	
			                    
                
				                on Feb 13, 2024 
				
			
			
		


				
		

		    		    
		    
		    	
			
And here is a new challenge! It's easier than the previous one but still...


 


Please spread those weekly quizzes among your colleagues!


Also, please be informed awards and goodies are being sent to the frequent winners. Please apologize for the long delay since we do that only 2 times per year!


 Certificates of the wins are also sent in paper form; the ones desiring a digital copy, please let us know!


Some statistics about the frequent winners by country origins: (we have also statistics by individuals, but they can be provided on demand and only with own data).


 


Regards


Kuo-Chang

									
			        
		


				
						
        

    
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                 equation
                
                                                    
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      8x+2x=130
      

      (23)x+2x=130
      

      (2x)3+2x=130
      

      Let a=2x which results in the cubic equation:
      

      a3+a-130=0
      

      By inspection, the real root is a=5.  The two other roots are non-real complex conjugate numbers.  See Cardano’s formula in Wikipedia for the calculation of the complex roots.
      

      X=log(a)/log(2)=2.321928095
      
			
      

		
			
	      

		
		
	
      
	
      
		
      
			
		
      
	
      
	
      
		
						
				
				
						
				
				
		
		
      
			
		
      
	
      

      

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      •  Analog Employees  
      
            
			                            
                                                                                                                        
            
				
			
            
												
						in reply to retiredEE
					
									
      
	
      

		
		
      
					
      Correct! That equation accepts 3 solutions: one real and 2 complex. Congratulations
      
			
      

		
			
	      

		
		
	
      
	
      
		
      
			
		
      
	
      
	
      
		
						
				
				
						
				
				
		
		
      
			
		
      
	
      

      

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						in reply to KCC
					
									
        
	
        

		
		
        
					
        If you calculate x from the complex solutions you express the numbers as |mag| times e^(iTheta).  Then you use the natural log (ln) to solve for x.  This results in an infinite number of answers separated by 2Pi.
        
			
        

		
			
	      

		
		
	
        
	
        
		
        
			
		
        
	
        
	
        
		
						
				
				
						
				
				
		
		
        
			
		
        
	
        

        

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						in reply to KCC
					
									
          
	
          

		
		
          
					
          If you calculate x from the complex solutions you express the numbers as |mag| times e^(iTheta).  Then you use the natural log (ln) to solve for x.  This results in an infinite number of answers separated by 2Pi.
          
			
          

		
			
	      

		
		
	
          
	
          
		
          
			
		
          
	
          
	
          
		
						
				
				
						
				
				
		
		
          
			
		
          
	
          

          

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