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The sum of a number and its reciprocal is $\frac{13}{6}$ Find the number.

 The sum of a number and its reciprocal is $\frac{13}{6}$ Find the number.

This Question has 1 answers.

Let the number be $x$.

According to the given condition:

$ x + \frac{1}{x} = \frac{13}{6} $

Multiplying both sides by $x$ to eliminate the fraction:

$ x^2 + 1 = \frac{13}{6} x $

Rearranging the equation:

$ 6x^2 - 13x + 6 = 0 $

Solving the quadratic equation $6x^2 - 13x + 6 = 0$ using the quadratic formula:

$ x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(6)(6)}}{2(6)} $

$ x = \frac{13 \pm \sqrt{169 - 144}}{12} $

$ x = \frac{13 \pm \sqrt{25}}{12} $

$ x = \frac{13 \pm 5}{12} $

Solving for $x$:

$ x = \frac{13 + 5}{12} = \frac{18}{12} = \frac{3}{2} $

$ x = \frac{13 - 5}{12} = \frac{8}{12} = \frac{2}{3} $

The two possible values of the number are $ \frac{3}{2} $ and $ \frac{2}{3} $.

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