Answer :

[tex]x+\frac{1}{x}=8\\\\\frac{x^2}{x}+\frac{1}{x}=8\\\\\frac{x^2+1}{x}=8\\\\x^2+1=8x\\\\x^2-8x+1=0[/tex]

[tex]a=1;\ b=-8;\ c=1\\\\\Delta=b^2-4ac\to\Delta=(-8)^2-4\cdot1\cdot1=64-4=60\\\\x_1=\frac{-b-\sqrt\Delta}{2a};\ x_2=\frac{-b+\sqrt\Delta}{2a}\\\\\sqrt\Delta=\sqrt{60}=\sqrt{4\cdot15}=2\sqrt{15}\\\\x_1=\frac{8-2\sqrt{15}}{2\cdot1}=4-\sqrt{15};\ x_2=\frac{8+2\sqrt{15}}{2\cdot1}=4+\sqrt{15}\\\\Answer:4-\sqrt{15}\ or\ 4+\sqrt{15}.[/tex]
AL2006
You have said that                 x + 1/x = 8

Multiply each side by 'x' :       x² + 1 = 8x

Subtract 8x from each side:    x² - 8x + 1 = 0

Solving this quadratic equation by means of the quadratic formula,
(the procedure is left as an exercise for the student), the two
solutions are:

x = 4 + √15 = 7.873  (rounded)
and
x = 4 - √15 = 0.127  (rounded)

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