Algebra

The real numbers x and y satisfy    
x^2 + y^2 - 8x + 6y + 23 = 0.    
Find the largest possible value of x + y.      

Given                               x² + y² – 8x + 6y + 23  =  0     

Group together               (x² – 8x     )  +  (y² + 6y     )  + 23  =  0    

                                       (x² – 8x     )  +  (y² + 6y     )           =  – 23   

Complete both squares and add the same    

amount on the right as you add on the left.    

                                         (x² – 8x + 16) + (y² + 6y + 9)  =  – 23  + 16  + 9     

                                         (x – 4)²  +  (y + 3)²  =  2    

This draws a circle centered at (+4, –3) with a radius the sqrt(2)     

We don't care about the center, all we care about is the radius.    

By formula, the largest x+y of a circle is its radius times the sqrt(2).    

So, the largest x+y is the sqrt(2) times the sqrt(2)         x + y = sqrt(2) • sqrt(2)    

                                                                                       largest x + y = 2    

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