Compute all values of x such that x-1 is the reciprocal of x+1. express your answer in simplified radical form.

x = ±√2

Step-by-step explanation:

         x - 1 =     1/(x + 1)

(x -1)(x + 1) =     1     Multiplied each side by x + 1

       x² - 1 =     1     Difference of two squares

           x² =     2     Add 1 to each side

           x  = ±√2

x-1 is the reciprocal of x+1 when x = -√2 or x = √2.

Check:

(a) x = -√2

  -√2 - 1 = 1/(-√2 + 1)

-(√2 + 1) = (√2 +1)/[(-√2 +1)(√2 + 1)]

-(√2 + 1) = (√2 +1)/(-2 +1)

-(√2 + 1) = (√2 +1)/(-1)

-(√2 + 1) = -(√2 +1)

(b) x = √2

√2 - 1 = 1/(√2 + 1

√2 - 1 = (√2 - 1)/[(√2 + 1)(√2 - 1) ]

√2 - 1 = (√2 - 1)/(2 - 1)

√2 - 1 = (√2 - 1)/1

√2 - 1 = √2 - 1

OK.

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