Let's simplify the given expression \((\sqrt{x-1} - \sqrt{x+1})^2\).

We can expand it using the formula for the square of a binomial: 

\[
(a - b)^2 = a^2 - 2ab + b^2
\]

In this case:
- \( a = \sqrt{x - 1} \)
- \( b = \sqrt{x + 1} \)

So, applying the formula:

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

Simplify each term:

1. \((\sqrt{x-1})^2 = x - 1\)
2. \((\sqrt{x+1})^2 = x + 1\)
3. \(2\cdot \sqrt{x-1} \cdot \sqrt{x+1} = 2\cdot\sqrt{(x-1)(x+1)} = 2\cdot\sqrt{x^2 - 1}\)

Now, substitute back into the expression:

\[
x - 1 - 2\sqrt{x^2 - 1} + x + 1
\]

Combine like terms:

\[
2x - 2\sqrt{x^2 - 1}
\]

Thus, the simplified expression is:

\[
2x - 2\sqrt{x^2 - 1}
\]

Let me know if you want more details or have any questions.

Here are five related questions:
1. How do you simplify other expressions involving square roots?
2. What is the geometric interpretation of square root functions?
3. How would this expression change if it involved cube roots?
4. How do you rationalize expressions with square roots in the denominator?
5. How can you apply binomial expansion to more complex terms?

**Tip:** When dealing with square roots, always check for restrictions on the domain (e.g., values that make the expression inside the square root negative).

Explore how to simplify the expression (\sqrt{x-1} - \sqrt{x+1})^2 step by step using the binomial expansion formula. Learn the key concepts of algebra, including handling square roots and combining like terms. This guide is suitable for students in Grades 9-11.

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