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 \( b = \left( 2 - \sqrt{3} \right) \). The expression for \( x \) is \( a^3 + b^{-3} \).We calculate \( a^3 \) and \( b^{-3} \) using binomial expansion. The expression \( a^3 + b^{-3} \) will simplify to a form suitable for direct substitution into the equation \( x^3 - 3x + k = 0 \).Upon simplification, we determine that \( k = 4 \). Therefore, the final answer is 2. 4.