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Suppose $a$ and $x$ satisfy $x^2 + \left(a-\frac{1}{a}\right)x  - 1 = 0$. Solve for $x$ in terms of $a$.

 Jan 22, 2018
 #1
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\($x^2 + \left(a-\frac{1}{a}\right)x - 1 = 0$\)

 

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

 

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

 

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

 

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

 

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

 

So  either       

 

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

 

Or

 

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

 

 

cool cool cool

 Jan 22, 2018

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