You are looking to solve the equation:

\[
\sqrt{y} - \frac{1}{\sqrt{y}} = 2
\]

We can try solving this step by step. Let's set:

\[
x = \sqrt{y}
\]

This makes the equation:

\[
x - \frac{1}{x} = 2
\]

Now, multiply both sides of the equation by \(x\) to eliminate the fraction:

\[
x^2 - 1 = 2x
\]

Rearrange the terms to form a quadratic equation:

\[
x^2 - 2x - 1 = 0
\]

Now solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 4}}{2}
\]

\[
x = \frac{2 \pm \sqrt{8}}{2}
\]

\[
x = \frac{2 \pm 2\sqrt{2}}{2}
\]

\[
x = 1 \pm \sqrt{2}
\]

So, \(x = 1 + \sqrt{2}\) or \(x = 1 - \sqrt{2}\).

Since \(x = \sqrt{y}\), and \(x\) must be positive, we discard \(x = 1 - \sqrt{2}\) (as it's negative). Hence, \(x = 1 + \sqrt{2}\).

Now, solve for \(y\):

\[
\sqrt{y} = 1 + \sqrt{2}
\]

Square both sides:

\[
y = (1 + \sqrt{2})^2
\]

\[
y = 1^2 + 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2
\]

\[
y = 1 + 2\sqrt{2} + 2
\]

\[
y = 3 + 2\sqrt{2}
\]

Thus, the value of \(y\) is:

\[
y = 3 + 2\sqrt{2}
\]

---

Would you like further details or have any other questions?

Here are some related questions to explore:
1. How do we apply the quadratic formula in different contexts?
2. What are the properties of square roots in algebraic equations?
3. How does squaring affect the sign of a number?
4. What other methods can we use to solve non-linear equations?
5. How does multiplying by the variable simplify fractional equations?

**Tip:** When simplifying quadratic equations, check for factorization before applying the quadratic formula—it might save you time!

This page provides a detailed solution to the algebraic equation sqrt(y) - 1/sqrt(y) = 2, exploring quadratic equations, the quadratic formula, and properties of square roots. Learn step-by-step how to solve for y. Suitable for students in Grades 9-12.

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