Question

If x² – √3x + 1 = 0, then find the value of x¹⁴ + x^-14.
1. -2
2. 1
3. 2
4. -1

Answer 1

Correct Answer - Option 2 : 1

Given:

x2 – √3x + 1 = 0

Formula used:

If x + 1/x = √3, then x⁶ = -1

If x + 1/x = k, then x² + 1/x² = k² - 2

Calculation:

x2 – √3x + 1 = 0

Dividing both side of the equation by x, we get

⇒ x - √3 + 1/x = 0

⇒ x + 1/x = √3

So, x⁶ = -1

Now, x14 + x-14.

⇒ x¹² × x² + x^-12 × x^-2 

⇒ (x⁶)² × x2 + (x⁶)-2 × x^-2

⇒ (-1)2 × x2 + (-1)-2 × x-2

⇒ x² + 1/x²

As x + 1/x = √3, 

Hence, x2 + 1/x² = (√3)² - 2

⇒ 3 - 2

⇒ 1

∴ The value of x14 + x-14 is 1.