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We are given the following system of equations:

\[
x + y = \sqrt{5}
\]
\[
xy = 2
\]

We are asked to find the expression:

\[
\frac{y}{x} + \frac{x}{y}
\]

### Step 1: Simplify the target expression

The expression \(\frac{y}{x} + \frac{x}{y}\) can be simplified using a common denominator:

\[
\frac{y}{x} + \frac{x}{y} = \frac{y^2 + x^2}{xy}
\]

We know that \(xy = 2\), so this becomes:

\[
\frac{x^2 + y^2}{2}
\]

### Step 2: Use the identity for \(x^2 + y^2\)

We can express \(x^2 + y^2\) using the identity:

\[
x^2 + y^2 = (x + y)^2 - 2xy
\]

From the given equations, we know \(x + y = \sqrt{5}\) and \(xy = 2\). Substituting these values:

\[
x^2 + y^2 = (\sqrt{5})^2 - 2(2) = 5 - 4 = 1
\]

### Step 3: Final calculation

Now, substitute \(x^2 + y^2 = 1\) into the earlier expression:

\[
\frac{x^2 + y^2}{2} = \frac{1}{2}
\]

Thus, the value of \(\frac{y}{x} + \frac{x}{y}\) is \(\frac{1}{2}\).

### Final Answer:
\[
\frac{y}{x} + \frac{x}{y} = \frac{1}{2}
\]

Would you like further details on any of the steps or additional examples?

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Here are 5 related questions to explore:
1. How would the solution change if \(x + y = 3\) and \(xy = 4\)?
2. What is the geometric meaning of the equations \(x + y = \sqrt{5}\) and \(xy = 2\)?
3. How can you solve quadratic equations derived from systems like these?
4. Can this approach be applied to find the roots of a quadratic equation?
5. How does this method relate to the concept of symmetric polynomials?

**Tip:** When dealing with symmetric expressions involving variables \(x\) and \(y\), use identities like \(x^2 + y^2 = (x + y)^2 - 2xy\) to simplify complex-looking expressions efficiently.

Si se satisfacen: x + y = √5, xy = 2, Hallar: y/x + x/y

Solve a system of equations where x + y = √5 and xy = 2, and find the expression y/x + x/y. This problem involves algebraic manipulation of symmetric expressions and quadratic identities.

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