If x2 – 9x + 1 = 0, what is the value of x8 – 6239x4 + 1?

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

then, $K^2+\frac{1}{K^2}$ = n² – 2

If x² – 9x + 1 = 0

what is the value of x⁸ – 6239x⁴ + 1

Divide x² – 9x + 1 = 0  both the sides by x we get,

x + \(\frac{1}{x}\) = 9

x² + \(\frac{1}{x^2}\) = 9² – 2 = 79

and,

x⁴ + \(\frac{1}{x^4}\) = 79² – 2 = 6239

x⁸ + 1 = 6239x⁴ 

Put this value in the required equation,

x⁸ – 6239x⁴ + 1 = x⁸ – x⁸ - 1 + 1 = 0