recriprocal

Hint:  Use Vieta's

Using the quadratic formula, we have the two solutions $\dfrac{13}{2} + \dfrac{3}{2}\sqrt{17}$ and $\dfrac{13}{2} - \dfrac{3}{2}\sqrt{17}$. The reciprocals would be $\dfrac{1}{\dfrac{13}{2} + \dfrac{3}{2}\sqrt{17}}$  and $\dfrac{1}{\dfrac{13}{2} - \dfrac{3}{2}\sqrt{17}}$. Adding them together, we have $\left(\frac{1}{\left(\frac{13}{2}+\frac{3}{2}\sqrt{17}\right)}\right)+\left(\frac{1}{\left(\frac{13}{2}-\frac{3}{2}\sqrt{17}\right)}\right) = \boxed{\dfrac{13}{4}}$.

- PM

The roots are

13 + 3√17                        13 - 3√17

_________     and          _________

     2                                       2

So

The reciprocal  sum is

      2                       2

_________  +    _________

13 + 3√17            13 - 3√17

  2 [ 13 - 3√17]  + 2 [ 13 + 2√13 ]

__________________________

        169  - 9(17)

  52              13

____  =       ___

  16               4

Thank you, everyone!