If $x^2 - 4x + 1 = 0$, then what is the value of $(x^6 + x^{-6})$?

2702

If $K+\frac{1}{K}=n$

then, $K^2+\frac{1}{K^2}$ = n² – 2

If x + \(\frac{1}{x}\)  = n

then, $x^3 +\frac{1}{x^3}$ = n³ - 3 × n

If $x^2 - 4x + 1 = 0$

Then what is the value of $(x^6 + x^{-6})$

Divide by x on both sides of $x^2 - 4x + 1 = 0$

x + \(\frac{1}{x}\) = 4

 $x^2+\frac{1}{x^2}$ = 4² – 2 = 14

Now cubing on both sides,

 $x^6+\frac{1}{x^6}$ = 14³ - 3 × 14

 $x^6+\frac{1}{x^6}$ =2744 - 42 = 2702