<s>[INST] Given the sum and the product of two variables; what's the sum of their reciprocals? I'm sorry if it's a simple problem, but I need an explanation..
If
$$ x + y = 2 $$
$$ xy = 3$$
Then:
$${1\over x} + {1\over y} = z  $$
$${z\in  \{ \Bbb R}\}$$
My attempt was as simple like that:
$$x=2-y$$
$$(2-y)y=3$$
$$y^2 -2y + 3 = 0$$
$$(y^2 -2y +1) +2 = 0$$
$$(y-1)^2 = -2$$
$$(y-1) = \pm \sqrt{-2} = \pm i\sqrt{2}$$
$$y= 1\pm i\sqrt{2}$$
$$x=2-(1\pm i\sqrt{2})$$
$$x= 1\pm i\sqrt{2}$$
Since $$x = y$$ Therefore:
$${1\over x} + {1\over y} = {2\over 1\pm i\sqrt{2}}$$
My question is about how to proceed after? or maybe I did a mistake before? How can I reach the final real solution?
Thank you.
 [/INST] Your approach is correct up to where you solved for the values of $y$ and $x$. The sum $\frac{1}{x} + \frac{1}{y}$ will always be equal to $\frac{1}{1 - i \sqrt 2} + \frac{1}{1 + i \sqrt 2}$. Cross multiplying gives you $\frac{1 + i \sqrt 2 + 1 - i \sqrt 2}{(1 - i \sqrt 2)(1 + i \sqrt 2)} = \frac{2}{1 + 2} = \frac{2}{3}$.
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