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If 23 = x^4 + x^4 + 2, what is the value of x^2 + 1/x^2?

 Mar 9, 2022
 #1
avatar+23198 
+1

x2 + 1/x2  =  (x4 + 1)/x2

 

23  =  x4 + x4 + 2     --->            23  =  (x4 + 1) + (x4 + 1)

                                --->            23  =  2(x4 + 1)

    divide by x2         --->        23/x2  =  2(x4 + 1)/x2         

                               --->     23/(2x2)  =  (x4 + 1)/x2 

                               --->           23/2  =  x4 + 1

                               --->           21/2  =  x4   

 

--->     x  =  fourth root of 21/2 

 Mar 9, 2022
 #2
avatar
+1

23 = x^4 + x^4 + 2

2 x^4 = 21

x^4 = 21/2

x^2 = sqrt (21/2)       ( taking only the positive root)

 

then    

sqrt (21/2) + 1/(sqrt 21/2) =   

[ (sqrt (21/2))^2 + 1]  / ( sqrt(21/2)) = 

23/2 / ( sqrt 21/2) =

23/2 (sqrt21/2) / (21/2) =

23/21 sqrt (21/2)                           or with further rationalization = 23/42 * sqrt (21)

 Mar 9, 2022
 #3
avatar+36433 
+2

or with further rationalization = 23/42 * sqrt (42)      *** edit ****

ElectricPavlov  Mar 9, 2022

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