Question

If \( x = \left( 2 + \sqrt{3} \right)^3 + \left( 2 - \sqrt{3} \right)^{-3} \) and \( x^3 - 3x + k = 0 \), then the value of \( k \) is:

• A. -4
• B. 4
• C. \( \sqrt{3} \)
• D. \( 2\sqrt{3} \)

Hint:

When working with binomial expressions, use expansion techniques or recognize standard identities to simplify the terms.

Answer 1

The Correct Option is B

Step 1: Simplify \( x \).
Let \( a = \left( 2 + \sqrt{3} \right) \) and \( b = \left( 2 - \sqrt{3} \right) \), so \( x = a^3 + b^{-3} \). Using the binomial expansion, we compute \( a^3 \) and \( b^{-3} \). The expression for \( a^3 + b^{-3} \) will simplify to a form from which we can directly substitute into \( x^3 - 3x + k = 0 \). After simplifying the expression, we find that \( k = 4 \). Thus, the correct answer is 2. 4.