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x is a real number such that x - 1/x = 3.  Find all possible values of x + 1/x.

 Jun 19, 2020
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\(x-1/x=3\\ x^2-1=3x\\ x^2-3x-1=0\\ x=\frac{3\pm\sqrt{13}}{2}\\ \)

So

 

x+1/x

=\(\frac{3+\sqrt{13}}{2}+\frac{1}{3+\frac{\sqrt{13}}{{2}}}\\ =\sqrt{13}\)

 

x+1\x

\(\frac{3-\sqrt{13}}{2}+\frac{1}{3-\frac{\sqrt{13}}{{2}}}\\ =-\sqrt{13}\)

 

So I believe the possible values of x+1/x are \(\boxed{\sqrt{13}\ \text{and}\ -\sqrt{13}.}\)

 Jun 19, 2020

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