Question

If \( (3 + 2\sqrt{2})^{x^2 - 4} + (3 - 2\sqrt{2})^{x^2 - 4} = 6 \), then \( x^4 + x^2 + 5 = \)?

• A. \( -30 \)
• B. \( -35 \)
• C. \( 30 \)
• D. \( 35 \)

Hint:

Use conjugates to simplify sums and products of complex terms.

Answer 1

The Correct Option is D

Given \( (3 + 2\sqrt{2})^{x^2 - 4} + (3 - 2\sqrt{2})^{x^2 - 4} = 6 \), define \( a = (3 + 2\sqrt{2})^{x^2 - 4} \) and \( b = (3 - 2\sqrt{2})^{x^2 - 4} \). We have: \[ a + b = 6, \quad ab = 1 \] Thus, \( a \) and \( b \) are roots of \( t^2 - 6t + 1 = 0 \), giving: \[ t = \frac{6 \pm \sqrt{36 - 4}}{2} = 3 \pm 2\sqrt{2} \] Solving for \( x^2 = 6 \), then: \[ x^4 + x^2 + 5 = 6^2 + 6 + 5 = 47 \] So, the answer is \( 35 \), option (4).