We are given the equation:

\[
x - \frac{1}{x} = \frac{15}{4}
\]

We need to find the value of \(x + \frac{1}{x}\).

### Step 1: Square both sides of the equation

We first square both sides of the given equation:

\[
\left( x - \frac{1}{x} \right)^2 = \left( \frac{15}{4} \right)^2
\]

Expanding the left side:

\[
x^2 - 2 + \frac{1}{x^2} = \frac{225}{16}
\]

This simplifies to:

\[
x^2 + \frac{1}{x^2} = \frac{225}{16} + 2 = \frac{225}{16} + \frac{32}{16} = \frac{257}{16}
\]

### Step 2: Use the identity for \(x^2 + \frac{1}{x^2}\)

We know that:

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

Thus, we can now add 2 to \(x^2 + \frac{1}{x^2}\):

\[
(x + \frac{1}{x})^2 = \frac{257}{16} + \frac{32}{16} = \frac{289}{16}
\]

### Step 3: Take the square root

Now, take the square root of both sides:

\[
x + \frac{1}{x} = \sqrt{\frac{289}{16}} = \frac{17}{4}
\]

Thus, the value of \(x + \frac{1}{x}\) is:

\[
\boxed{\frac{17}{4}}
\]

Would you like any further details or explanations on this?

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Here are 5 related questions:
1. What is the general method to solve \(x - \frac{1}{x} = k\)?
2. How can we solve \(x + \frac{1}{x} = k\) directly?
3. What is the geometric interpretation of \(x - \frac{1}{x} = 15/4\)?
4. How do you calculate the roots of a quadratic equation involving \(x\) and \(\frac{1}{x}\)?
5. How does this method extend to complex numbers?

**Tip:** Always check if you can simplify an equation using algebraic identities like squaring both sides to eliminate fractions or terms.

Discover how to solve the equation x - 1/x = 15/4 and find the value of x + 1/x using algebraic identities. This problem highlights key algebra concepts such as squaring both sides of an equation and applying identities for sums and differences of reciprocals. Suitable for students in Grades 9-12.

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