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Suppose a and x satisfy x^2 + (a - (1/a))x - 1 = 0. Solve for x in terms of a.

 Jun 9, 2019
 #1
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x^2 + (a - (1/a))x - 1 = 0

 

x^2 +[ (a^2 - 1) / a ]x - 1  = 0   complete the square on x

 

x^2 +  [ (a^2 - 1) / a ] x  =  1

 

x^2 + [ (a^2 - 1) / a ] x  +  (a^2 - 1)^2/ [4a^2]  =   1 + (a^2 - 1)^2 / [4a^2 ]

 

(x + (a^2 - 1)/[2a] )^2   = [ 4a^2 + a^4 - 2a^2 + 1 ] / [ 4a^2]

 

(x + (a^2 - 1) / [2a] )^2  =  [ a^4 + 2a^2 + 1] / [ 4a^2]

 

(x + (a^2 - 1) / [2a])^2 =  [ a^2 + 1]^2 / [ 4a]^2      take both roots

 

( x + (a^2 - 1) / [2a] )  =  ±√ [( a^2 + 1)^2 / [4a]^2]

 

x + (a^2-1) / [2a]  = ± ( [ a^2 + 1] / [2a] )

 

x =  ± ([a^2 + 1] / [2a]) - (a^2-1) /[2a ]

 

 

cool cool cool

 
 Jun 9, 2019

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