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 Find the largest real number x for which there exists a real number y such that x^2 + y^2 = 2x + y.

 Nov 30, 2020
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x^2  + y^2   = 2x  + y

 

x^2 - 2x  + y^2  - y   =   0      complete the square on  x and y

 

x^2  - 2x + 1  + y^2  - y  + 1/4  =   1 + 1/4      factor  

 

(x - 1)^2  +  ( y - 1/2)^2  =   5/4

 

This is a circle  with a center of   (1, 1/2)   and a radius of  sqrt  (5/4)  =  sqrt (5)  / 2

 

So.....the largest x that will provide a real numer for y  is   when  x =  (1 + sqrt (5)/2) 

 

cool cool cool

 Nov 30, 2020

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