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There are two pairs (x,y) of real numbers that satisfy the equation x + y = 3, xy = 1. Given that the solutions x are in the form x = (a pm b*sqrt(c))/d where a, b, c, and d are positive integers and the expression is completely simplified, what is the value of a + b + c+ d ?

 Jul 15, 2021
 #1
avatar+121008 
+2

x +  y =  3      

xy   = 1   ⇒  y  = 1/x

 

So

 

x +  1/x  = 3        multiply through  by   x

 

x^2  + 1  =   3x

 

x^2  - 3x    =   -1             complete the  square on x

 

x^2  - 3x  + 9/4  =  -1  +  9/4

 

(x - 3/2)^2   =  5/4         take both roots

 

x - 3/2  =  -sqrt (5/4)           or       x   - 3/2  = sqrt (5/4)

 

x  - 3/2  =   -sqrt (5) / 2                    or       x  - 3/2  =  sqrt (5) /2

 

x =    (3 -sqrt (5)) / 2                                   x =  (3 + sqrt (5) ) / 2

 

a   =  3         b = 1        c = 5      d = 2

 

Sum =   11

 

 

cool cool cool

 Jul 15, 2021
 #2
avatar+17 
+2

Wow @CPhill

 

That solution was brilliant!

cool

 Jul 15, 2021
 #3
avatar+121008 
+1

LOL!!!!........Thx  !!!

 

 

cool cool cool

CPhill  Jul 15, 2021

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