5 and 1/5.

Those are the answers. I hope you solved it the easy way.

No, I can’t. I might manage it on paper.

The easy way would be to plug it into Google. I did it in my head.

For me, if it’s rational I put it into a ratio. I got to the equation (x^^2+1)/x = 26/5 [x^^2 means x squared] and saw that if the denominators match (x = 5) then so will the numerators. Then, going by the original equation, which tells us it’s the sum of a number and its reciprocal, I knew 1/5 must be the other answer.

I tried working with decimals, too. I got to the polynomial x^^2 – 5.2x + 1 = 0. From here I admit I summoned the integers for a moment and used 10x^^2 – 52x + 10 to help me factor it into (x – 5)(x – 0.2). Integer coefficients helped me find 50 and 2 as opposed to getting 5 and 0.2.

The first time I solved it, I’d imagined parentheses around (x + 1). It was simple to step through in my head, but your hint hadn’t helped me much (haha, that should have been my clue).

Then I saw @JeSuisRickSpringfield‘s response, looked back at the question, and realized I’d misread the problem.

And then your hint made sense: x + 1/x = 5 + 1/5

I’m pretty sure I would have gotten 5 without having seen @JeSuisRickSpringfield‘s response, but I’m also pretty sure I would have missed 1/5. (Having seen it as a solution, though, I see how it works.)

I’ll admit it. I immediately stared to solve for it in my head before I saw your hint. I multiplied everything by x and made a polynomial. Then i saw the hint. That clinched it.

I did it in my head. The .2 was a dead giveaway that X=5