Question : If $x^2-4 x+1=0$, then what is the value of $\left(x^6+x^{-6}\right)$?
Option 1: 2786
Option 2: 2702
Option 3: 2716
Option 4: 2744
Correct Answer: 2702
Solution :
Given: $x^2 - 4x + 1 = 0$
Divided by $x$
⇒ $x - 4 + \frac{1}{x} = 0$
⇒ $x + \frac{1}{x} = 4$
As we know,
$(x + \frac{1}{x})^2 = x^2 + \frac{1}{x^2} + 2$
⇒ $4^2 = x^2 + \frac{1}{x^2} + 2$
⇒ $x^2 + \frac{1}{x^2}$ = 16 – 2 = 14
Again as we know,
$(x^2 + \frac{1}{x^2})^3 = x^6 + \frac{1}{x^6} + 3(x^2 + \frac{1}{x^2})$
⇒ $14^3 = x^6 + \frac{1}{x^6} + 3 × 14$
⇒ $x^6 + x^{-6}$ = 2744 – 42 = 2702
Hence, the correct answer is 2702.
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