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I found the root of this equation is: $ x = \frac{1}{3} (1 \pm \sqrt{13}) $. How can I convert this result to fraction?. Sorry for my ignorance, I don't practice math for a long time.

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@WillJagy: mind explaining your link a bit? – Willie Wong Dec 16 '11 at 9:21
@WillieWong the user has changed his icon. Yesterday he had an "ambigram," the word "earth" with a rotational 180 degree symmetry, commissioned by Dan Brown for his book Angels and Demons. A key feature, as in those captcha tests, is our willingness and ability to extend the alphabet a bit. In the case of ambigrams we gather together regions that are not topologically connected into a letter. – Will Jagy Dec 16 '11 at 20:15
up vote 5 down vote accepted

You can show it as $\frac 13 \pm \frac{\sqrt{13}}{3}$ or $\frac{1 \pm \sqrt{13}}{3}$. Does either of these meet your needs? You will not get rid of the square root sign-these numbers are not rational.

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One more comment: if it were the case that the solution was a rational number, then the numerator would have to be a divisor of 4 (if you have never seen this, you can try to prove it, or look at this Wikipedia entry). But if you could write $1\pm\sqrt{13}=d$, where $d\in\{\pm 1,\pm2,\pm4\}$, then this would contradict the irrationality of $\sqrt{13}$.

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You don't need the rational root test if you know that $\sqrt{13}$ is irrational: If $\frac{1\pm\sqrt{13}}{3}=q$ were rational, then $\sqrt{13} = \mp(3q-1)$ would be rational. – Myself Dec 15 '11 at 19:37

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