Given that \( x = 2 - \sqrt{3} \), we need to find the value of \( x^2 - \frac{1}{x^2} \).

### Step 1: Find \( x^2 \)
We begin by squaring both sides of the equation \( x = 2 - \sqrt{3} \).

\[
x^2 = (2 - \sqrt{3})^2
\]

Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):

\[
x^2 = 2^2 - 2(2)(\sqrt{3}) + (\sqrt{3})^2
\]
\[
x^2 = 4 - 4\sqrt{3} + 3
\]
\[
x^2 = 7 - 4\sqrt{3}
\]

### Step 2: Find \( \frac{1}{x} \)
Now, we need to compute \( \frac{1}{x} \). Since \( x = 2 - \sqrt{3} \), we rationalize the denominator by multiplying by the conjugate:

\[
\frac{1}{x} = \frac{1}{2 - \sqrt{3}} \times \frac{2 + \sqrt{3}}{2 + \sqrt{3}}
\]
\[
\frac{1}{x} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3})(2 + \sqrt{3})}
\]
Using the difference of squares:

\[
\frac{1}{x} = \frac{2 + \sqrt{3}}{4 - 3}
\]
\[
\frac{1}{x} = 2 + \sqrt{3}
\]

### Step 3: Find \( \frac{1}{x^2} \)
Square \( \frac{1}{x} \):

\[
\frac{1}{x^2} = (2 + \sqrt{3})^2
\]
Using the same expansion formula:

\[
\frac{1}{x^2} = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2
\]
\[
\frac{1}{x^2} = 4 + 4\sqrt{3} + 3
\]
\[
\frac{1}{x^2} = 7 + 4\sqrt{3}
\]

### Step 4: Calculate \( x^2 - \frac{1}{x^2} \)
Now we subtract \( \frac{1}{x^2} \) from \( x^2 \):

\[
x^2 - \frac{1}{x^2} = (7 - 4\sqrt{3}) - (7 + 4\sqrt{3})
\]
\[
x^2 - \frac{1}{x^2} = 7 - 7 - 4\sqrt{3} - 4\sqrt{3}
\]
\[
x^2 - \frac{1}{x^2} = -8\sqrt{3}
\]

Thus, the value of \( x^2 - \frac{1}{x^2} \) is \( -8\sqrt{3} \).

Would you like further details or have any questions?

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Here are 5 related questions for practice:
1. If \( x = 3 - \sqrt{2} \), find \( x^2 - \frac{1}{x^2} \).
2. Simplify \( (a - b)^2 \) and explain the process.
3. What is the value of \( x^4 - \frac{1}{x^4} \) for \( x = 2 - \sqrt{3} \)?
4. Explain the process of rationalizing the denominator of \( \frac{1}{a - b} \).
5. What is the general formula for \( x^n - \frac{1}{x^n} \) if \( x = a - b \)?

**Tip:** When rationalizing a denominator, always multiply by the conjugate to eliminate square roots.

Explore a step-by-step solution to finding x² - 1/x² when x = 2 - √3. This problem involves key algebraic concepts such as binomial expansion, rationalization of radicals, and difference of squares. Suitable for students in Grades 10-12.

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